Modified Pseudo-Inverse Method with Generalized Linear Quadratic Regulator for Fault Tolerant Model Matching with Prescribed Stability Degree

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1 th IEEE Conerence on Decision and Control and European Control Conerence (CDC-ECC) Orlando, FL, USA, December 12-15, 2011 Modiied Pseudo-Inverse Method with Generalized Linear Quadratic Regulator or Fault Tolerant Model Matching with Prescribed Stability Degree Bogdan D. Ciubotaru, Marcel Staroswiecki and Nicolai D. Christov Abstract When aults turn the nominal operation to an unstable post-ault regime, the accommodation procedure must recover stability and as much perormance o the reerence model as possible, ideally attaining perect Model Matching (MM). In this paper, the authors develop the Modiied Pseudo- Inverse Method (M) with the Generalized Linear Quadratic Regulator (GLQR) stabilization using the weighting matrices derived rom the Robust Optimal Model Matching (ROMM) approach and show that the M-GLQR technique is the best candidate to solve the MM problem extended with the prescribed stability degree property, as opposed to the classical M with simple LQR stabilization using arbitrary penalties, as was proposed in the original method. The theoretical development presented in the paper is illustrated by an aeronautical example, namely the longitudinal motion control o the B747 aircrat impaired by a structural and an actuator ault. I. INTRODUCTION Fault Tolerant Control (FTC) reers to the approach by which the controlled system is able to exhibit desired properties, e.g., Model Matching / Model Following (MM/MF), both in normal operation and in the presence o aults [1], [2], [3]. Model Matching assumes that the closed-loop matrix o the actual system is identical with the closed-loop one o the reerence model, while in Model Following the identity should hold or the state trajectories, in both cases, the matching / ollowing conditions being mathematically expressed by zeroing the corresponding matrix- / vector- norms o the error [4], [5]. Unortunately, MM/MF stresses an idealistic objective in relation with the FTC design, because achieving exactly the same closed-loop perormance as in the nominal case ater the system impairment is most oten impossible and approximate quality has to be accepted [6]. In this paper, the Modiied Pseudo-Inverse Method (M) [4] with the Generalized Linear Quadratic Regulator (GLQR) stabilization using the weighting matrices derived rom the Robust Optimal Model Matching (ROMM) approach [7] is developed. B.D. Ciubotaru is with APCC, Automatic Process Control and Computers Laboratory, Polytechnic University o Bucharest, Bucharest, Romania bogdancd@indin.pub.ro M. Staroswiecki is with SATIE, Systèmes et Applications des Technologies de l Inormation et de l Énergie Laboratory, École Normale Supérieure de Cachan, Cachan Cedex, France marcel.staroswiecki@univ-lille1.r N.D. Christov is with LAGIS, Laboratoire d Automatique, Génie Inormatique et Signal, Université des Sciences et Technologies de Lille, Villeneuve d Ascq Cedex, France nicolai.christov@univ-lille1.r The authors show that the M-GLQR technique is the only pertinent approach to solving the MM problem extended with the Prescribed Degree o Stability (PDS) property, contrary to the classical M with simple LQR stabilization using arbitrary penalties, as suggested in the original M- LQR method. Moreover, two ault accommodation schemes under the orm o the M-G/LQR algorithms are proposed - also, the authors improve at the theoretical level and organize better the previous version o the paper [8], and provide it with an application example rom aeronautics or the numerical and graphical illustration o the advantage o using M-GLQR to solving MM-PDS, contrary to the M- LQR which ails. The paper is organized as ollows: Section II addresses the perect MM problem with the Pseudo-Inverse Method () solution, recalling its instability drawback, while Section III tackles the approximate MM problem with the Modiied Pseudo-Inverse Method (M), recalling its inability to satisy the PDS property and proposes the M-GLQR solution. Also, an application example or the ault tolerant model matching o the B747 longitudinal motion control is provided in Section IV and the paper is concluded in Section V. In the end, Appendix I provides the reader with the particular solution given by the ROMM procedure. II. PSEUDO-INVERSE MODEL MATCHING A. Nominal Operation Consider the continuous linear time-invariant (LTI) deterministic system whose nominal operation is modeled by M n : ẋ(t) = A n x(t)+b n u n(t), with x(t 0 = 0) = x 0, (1) where A n R n n and B n R n m are the nominal system and control matrices, x(t) R n and u n (t) Rm are the state and control vectors, and x 0 R n is the initial condition. In nominal operation, the linear state-eedback control law conducts to the closed-loop system u n(t) = K nx(t), (2) ˆM n : ẋ(t) =  nx(t), with  n = A n B n K n, (3) which is stable and provides the ideal dynamic perormance, where by αn λ(â n ) the maximum absolute value o the real-part o the eigenvalues o  n is denoted /11/$ IEEE 1583

2 Model ˆM n is taken as the reerence model whose dynamics is to be matched in any other operating conditions dierent rom the ideal one, while α n indicates the minimum distance imposed between the eigenvalues position to the imaginary axis in the complex plane, namely the Prescribed Degree o Stability (PDS) [9]. B. Post-Fault Operation Whether parametric aults are considered, assuming that an LTI model is still able to capture its behavior, the system to be controlled is described by the pair (A,B ) provided by an on-line Fault Detection, Isolation, and Identiication / Estimation (FDIE) module [10]. That is, ater the ault estimation time t, the post-ault system operation is modeled by M : ẋ(t) = A x(t)+b u (t). (4) In post-ault operation, the Exact Model Matching (EMM) technique with state-eedback aims at designing the linear control law u (t) = K x(t), (5) such that the resulting closed-loop behavior matches that o the reerence model rom (3) (ull state is assumed to be available and known ater the ault - otherwise, state estimation must be perormed beore controller redesign). The solution to the EMM problem is obtained by solving the matrix equation  A B K =  n, (6) whose necessary and suicient condition or a solution to exist is expressed as (see [11]) and the solution is given by Im ( A  n) Im ( B ), (7) (K )K = B (A  n), (8) where (.) stands or the pseudo-inverse matrix. C. Pseudo-Inverse Method Regarding the pseudo-inverse control (8), condition (7) will obviously hold only or very particular aults and thereore no exact solution will exist in most ault cases; or this reason, approximate solutions rather than exact ones might be o interest. When exact model matching is impossible (   n), an approximate control solution may be computed as K = arg min K J (K ) by minimizing the criterion (see [4]) with the impairment matrix J (K ) = Ê F, (9) Ê =  n (A B K ) (  =  n Ê ), (10) where. F is the Frobenius matrix-norm. Unortunately, or certain ault cases, encounters instability problems (see also [12]); besides that, using this approach, no action to make the prescribed degree o stability tangible can be taken. III. MODIFIED PSEUDO-INVERSE MODEL MATCHING A. Modiied Pseudo-Inverse Method without Stabilization and without Correction The improved, namely the Modiied Pseudo-Inverse Method (M), counteracts the above instability drawback, that is it restricts the computation o K such that to guarantee the stability o the post-ault system Λ( ) C while achieving as much o the closed-loop nominal perormance as possible, i.e., min K J (K ), where Λ(.) denotes the spectrum o the matrix-argument. Unortunately, there is no closed-orm solution working or general multi-input multi-output (MIMO) systems but just or the single-input multi-output (SIMO) case, that is in the case when m = 1. This is the situation that will be tackled in the paper - though B R n 1, it still is appropriate or the control o the longitudinal motion o an aircrat, classically done using just the elevator input, which will be the application example. Thereore, in M, the control gain K M (or j = 1,2,...,n) computed as k M k M = k = sgn ( k, i k δ, (11a) ) δ, otherwise, (11b) solves the reconigurable control problem with stability guarantees, where δ represents the truncated robust stabilitymargin, computed as δ = δ ε, or some small ε, and sgn(.) stands or the sign-unction o the realargument (or details on the computation o δ, and o δ -G/LQR used later in the next subsection, see below in this subsection). This comes rom the act that i the post-ault closed-loop system designed with, described as ẋ(t) =  x(t) = (A B K )x(t), is unstable, then the pseudo-inverse control must be redesigned with the modiied-pseudo inverse method such that the M closed-loop system, described as ẋ(t) =  M x(t) = (A B K M )x(t) = (A δ B K )x(t), becomes stable, rom where K M = (K ). The idea is to see the closed-loop matrix in the post-ault case  M A E M as a perturbed matrix  M = A B K M, where the perturbation matrix, denoted by E M, is taken as E M assumed to be bounded by δ that K M = δ K. B K M = B (K ), and, rom where it will ollow Thus, choosing the stability bound to be δ = δ Y [13], the right-hand side o (12), k < 1 sup ω 0 ρ((jωi n A ) 1 E ) δ Y, or j = 1,2,...,n, (12) 1584

3 which involves the evaluation o the resolvent-matrix at the limit but the bound is less restrictive than those proposed in other results [14] (see also [15]), where ρ(x) denotes the spectral-radius o the matrix-argument, precisely the magnitude o the largest eigenvalue o X, condition k M δ, or j = 1,2,...,n, (13) guarantees the stability o  in the impaired operation. B. Modiied Pseudo-Inverse Method with Stabilization and with Correction Unortunately, one o the deiciencies o M is that the robust stability margin cannot be always used but in the case in which matrix A remains stable ater the ault, i.e., Λ(A ) C, otherwise the system must irst be stabilized using another method. 1) Fault Accommodation Control with LQR Stabilization and M Correction: Hence, or certain instability cases, in the classical paper on M [4], it was suggested to use irst an eicient method to get the stabilizing gain, e.g., through solving the corresponding Continuous Algebraic Riccati Equation (CARE) associated with the post-ault pair (A,B ), that is CARE : (14) A T XCARE + X CARE A X CARE B R 1 B T XCARE + Q = 0, where the solution to CARE is computed as K CARE and then to make a control correction as K M-LQR = K CARE + = R 1 B T XCARE, (15) -LQR δ K -LQR, (16) where the control accommodation dierence K -LQR is obtained as K -LQR = B (ÂLQR  ) n = B ( (A B K CARE )  n) ; (17) as no restrictions are indicated in that paper, see that the penalty matrices Q T = Q = Q T Q 0 and R T = R > 0 might be chosen as in (32a)-(32b). For this situation, the ault accommodation control redesign scheme with simple LQR stabilization and M correction becomes a two step procedure as ollows [4]. Algorithm 1: M-LQR Scheme (1-1) Stabilize the post-ault system using the classical LQR procedure o (14) with the weighting matrices Q,R as in (32a)-(32b), provide K CARE rom (15). (1-2) Make the appropriate control correction using the dierence matrix K -LQR as in (17), provide K M-LQR rom (16). However, again, look that there is no touch onto the imposed degree o stability using this approach. 2) Fault Accommodation Control with GLQR Stabilization and M Correction: Contrary to the simple LQR stabilization rom the original approach and with direct regard on the model matching problem with prescribed stability degree, in this paper, the authors propose to obtain the initial stabilizing post-ault control matrix through solving the Generalized Continuous Algebraic Riccati Equation (GCARE) (similar to (30)) associated with the aulty pair (A,B ), that is GCARE : (18) à T X GCARE + X GCARE à X GCARE B R 1 B T X GCARE + Q = 0, where the solution to GCARE is computed as K GCARE and then to make a control correction as K M-GLQR = R 1 (B T X GCARE + S ), (19) = K GCARE + -GLQR δ K -GLQR, (20) where the control accommodation dierence K -GLQR is obtained as K -GLQR = B (ÂGLQR  ) n = B ( (A B K GCARE )  n), (21) again ater an appropriate choice o the penalty matrices Q,R,S as in (32) (or details on the computation o Q,R,S, see Appendix I, Review on Robust Optimal Model Matching). For this situation, the ault accommodation control redesign scheme with generalized GLQR stabilization and M correction becomes a two step procedure as ollows. Algorithm 2: M-GLQR Scheme (2-1) Stabilize the post-ault system using the generalized GLQR procedure o (18) with the weighting matrices Q,R,S as in (32), provide K GCARE rom (19). (2-2) Make the appropriate control correction using the dierence matrix K -GLQR as in (21), provide K M-GLQR rom (20). From Algorithms 1 and 2, it appears clear now that only ollowing the steps o the second one, namely M- GLQR, the model matching problem with the imposition o prescribed stability degree is solved, as this property is inherently assured by the GLQR stabilization and it is not aected by the M correction - this is also to be proven numerically in the case study or the impaired model o the aircrat longitudinal motion. C. Interpretation o M-G/LQR Control Actions and Future Research The authors have showed in [8] that, ater making use o the weighting matrices Q,R rom (32a)-(32b) or the simple LQR stabilization respectively o Q,R,S rom (32) or the generalized GLQR stabilization, the -LQR and the -GLQR control accommodation matrices K -LQR and K -GLQR rom (17) and (21) get special orms. 1585

4 Following that development, the corresponding control matrices become K M-LQR K M-GLQR = δ -LQR = K +(1 -LQR δ )K CARE, (22) -GLQR δ )K CARE, (23) K +(1 where the M-LQR control provides to have somehow a bi-objective orm, and remark the clear dierence between the two which is proved through the act that in the M- GLQR control the component is penalized no more. -G/LQR Moreover, see that i the stability-bounds δ are much smaller than unity, their inluence becomes negligible, such that (22) and (23) get the approximate orms K M-LQR K M-GLQR K CARE, (24) K + K CARE ; (25) -G/LQR in any case, having δ 1 is not always the case. Also, i the ault is severe and the ideal perormance imposed through  n is too demanding, the control eort spent or stabilization will conduct to such K CARE that might be several orders o magnitude bigger than K in norm, which will imply in (25) that the correction becomes insigniicant regarding the GLQR stabilization, that is K M-GLQR K M-LQR ; (26) o course, this is a too qualitative analysis and urther investigation on the magnitude o the corresponding Frobeniusnorms o both control accommodation matrices must be done. In the sequel, the design o a controller or the longitudinal model o the B747 civil aircrat is considered, with the aim o proving the advantages o using the modiied pseudo-inverse method with generalized linear quadratic stabilization and post-ault penalty matrices provided by the robust optimal model matching approach or better ault tolerant model matching perormance with prescribed stability degree over the method with simple linear quadratic stabilization, ater a structural and an actuator ault. IV. APPLICATION OF M-G/LQR CONTROLS FOR THE B747 AIRCRAFT LONGITUDINAL MODEL A. Flight Condition Consider an aircrat Boeing 747 in gradual maneuvering without precision tracking (Class I cruise light, Category B light phase) and assume a straight and constant level light at t ixed altitude and 0.8 Mach - the corresponding steady-state speed is 774 ps; the aircrat mass is slug. B. Nominal System 1) Open-Loop Model: The aircrat longitudinal dynamics is described by the state vector x = [ u w q θ ] T, where u ( t/sec) and w ( t/sec) represent the inertial velocities in the x- and z- directions o F B reerence rame; (F B denotes body-axis reerence); also, q (rad/sec) and θ (rad) represent respectively the pitch rate and the pitch angle. The control input δ E (rad) is the elevator delection. The linearized model o the system is given by the nominal pair o system and control matrices (A n,b n ) as (see [16], or the aircrat longitudinal model) A n = B n = [ ] T., 2) Closed-Loop Reerence Model: Consider the reerence model  n(  n ) as  n = , computed with the ull state-eedback control gain K n = [ ], which gives the stable spectrum Λ( n) = { ± ı, ± ı}, or the nominal closed-loop matrix  n = A n B n K n. The reerence model  n provides ideal dynamic perormance or the natural modes o the longitudinal motion o the aircrat, precisely or the short-period (SP) mode eigenvalues λ1,2 SP = ± ı this provides the pair o natural undamped requency (rad/sec) and damping coeicient (ω SP,ζ SP ) (3,0.6) and or the phugoid (PH) mode eigenvalues λ1,2 PH = ± ı this gives the pair (ω PH,ζ PH ) (0.1,0.6) [17]. C. Impaired System 1) Post-Fault Model: Consider an actuator ault deined by the loss-o-eectiveness with 50% in the elevator delection; that can be modeled in the post-ault system by multiplying the control matrix with a actor (1 τ ) = 0.5, or τ = 0.5, i.e., B = (1 τ )B n. Also, assume a structural ault modeled in the system matrix as A = A n + A I 4, with A = σn max 10 3, where σn max = is the maximum singular value o the nominal matrix A n. Furthermore, consider also that the actual post-ault matrices provided by the FDIE module are A = (1 γ )A and B = (1 γ )B, with γ = 0.1, thus denoting a 10% structured identiication error (see [18], or the aircrat ault models). 2) Pseudo-Inverse Solution: The Pseudo-Inverse Method () provides the control gain K = [ ], which is unacceptable, as it gives an unstable spectrum Λ( ) = { ± ı, ± ı}, or the closed-loop matrix  = A B K. 1586

5 3) Modiied Pseudo-Inverse Solution: The Modiied Pseudo-Inverse Method (M) provides the control gain K M = [ ], which still is unacceptable, as it is not able to provide a stable spectrum Λ( M ) = { ± ı, ± ı}, or the closed-loop matrix  M = A B K M ; this is due to the act that matrix A is unstable, with the spectrum Λ(A ) = { ± ı, ± ı}. D. Accommodated System 1) Post-Fault Weighting Matrices: The penalty matrices derived as in the Robust Optimal Model Matching procedure, described in Appendix I, are identiied as Q = R = , , (27a) (27b) S = [ ] T, (27c) and the minimal decay rate α = 2.0 is chosen according to its deinition, precisely a little bit smaller than αn = ) M-LQR Solution: The simple LQR stabilization, using just the two penalties Q,R rom (27a)-(27b), with M correction, the M-LQR technique, provides the control gain K M-LQR = [ ], which still is not acceptable, as it provides the stable spectrum Λ( M-LQR )={ , , ±0.1858ı}, or the closed-loop matrix  M-LQR = A B K M-LQR, but it does not manage to obey the superior limit or the negative values o the real-parts o the eigenvalues in its spectrum, i.e., ) M-GLQR Solution: The generalized GLQR stabilization, using all the three penalties Q,R,S rom (27), with M correction, the M-GLQR technique, provides the control gain K M-GLQR = [ ] 10 5, which is inally acceptable, as it provides the stable spectrum Λ( M-GLQR )={ , , ±3.6603ı}, or the closed-loop matrix  M-GLQR = A B K M-GLQR, being the only approach that stabilizes the system and achieves the prescribed stability degree w (t/sec) q (rad/sec) SP vertical inertial velocity 60 ideal 80 M GLQR M LQR SP pitch rate Time (sec) Fig. 1. M-G/LQR Accommodation o the B747-SP Mode 4) M-GLQR vs. M-LQR Control: Unortunately, both ault tolerant model matching techniques provide large values or the dierences between the corresponding closedloop matrices and the reerence model one, but this is to be explained by the act that both generalized / classical quadratic regulators provide large control gains. However, there is one major advantage o the M- GLQR approach over the M-LQR in that it provides realparts o the closed-loop eigenvalues smaller than α, which makes M-GLQR the best candidate i, in the post-ault operation, the prescribed degree o stability is also imposed or the perormance to be ollowed as well, as a plus to the primary minimum Frobenius-norm requirement. 5) Closed-Loop Simulation: Due to the limited space in the paper, only the simulations or the short-period (SP) mode behavior (vertical inertial velocity and pitch rate) when applying the M-G/LQR accommodating controls are shown in Fig. 1, during the post-ault period (restricted on the graphs to t = 0). At least rom the graph o the pitch rate, there is to be seen that the M-GLQR is a better ollower than M-LQR. V. CONCLUSION This paper interrogated the Modiied Pseudo-Inverse Method (M) used as an accommodation technique that recovers as much perormance as beore the aults in the situation o impairments turning the ideal nominal operation to an unstable post-ault regime. The authors have developed the M with the Generalized Linear Quadratic Regulator (GLQR) stabilization using the penalty matrices derived rom the Robust Optimal Model Matching (ROMM) procedure and have shown that this is the only successul approach or the model matching problem extended with the prescribed stability degree property, as opposed to the classical M used in conjuction with simple LQR stabilization.

6 Moreover, an application example or the ault tolerant model matching o the B747 longitudinal motion control is also provided. APPENDIX I REVIEW ON ROBUST OPTIMAL MODEL MATCHING Let  n be the ideal nominal closed-loop matrix rom (3) and denote by αn λ(â n ) α the maximum absolute value o the real-part o the eigenvalues o  n. One o the solutions to the model matching problem with prescribed stability degree rom (1)-(3) known in the literature, with some robustness and optimality properties attached, designs the post-ault state-eedback so to minimize the perormance index (see [7]) J (α,â n) = e 2α t ( ẋ(t)  nx(t)  nx(t) 2 ) 2 dt ; t (28) the exponent α αn is deined as the desired minimal decay-rate o the post-ault closed-loop system, namely the exponential convergence rate o system state to the equilibrium point [19]. In this case, the optimal control u (t) = K x(t) which minimizes J (α,â n) rom (28) subject to (A,B ) is computed as K = R 1 (B T X + S ), (29) where the unique stabilizing symmetric positive-semideinite matrix X represents the solution to the Generalized Continuous Algebraic Riccati Equation (GCARE) GCARE : à T X + X à X B R 1 B T X + Q = 0, (30) with the buer matrices à = A + α I n B R 1 S, Q = Q S T R 1 S, (31) where the required post-ault weighting matrices rom (30)- (31), are identiied as (see also [20]) Q = A T A A T  n  T n R = B T B, S = B T (A  n ), A + 2 T n  n, (32a) (32b) (32c) obtained ater the reduction o the perormance integral in (28) to the corresponding generalized orm J (α )= e 2α t ( x(t) 2 Q + u (t) 2 R + 2u (t) T S x(t) ) dt, t (33) or the squared Q - and R - state- and control- vectors norms, i.e., x(t) 2 Q = x(t) T Q x(t) and u (t) 2 R = u (t) T R u (t), and S the control-state coupling matrix. ACKNOWLEDGMENTS The work o the irst author has been co-unded by the Sectorial Operational Programme Human Resources Development o the Romanian Ministry o Labor, Family and Social Protection through the Financial Agreement POSDRU/89/1.5/S/ The authors also acknowledge the support o this work by the ARCUS international research project unded by the French Ministry o Foreign Aairs and Nord - Pas-de-Calais Region. REFERENCES [1] J. Jiang, Fault tolerant control systems - An introductory overview, Acta Automatica Sinica, vol. 31, no. 1, pp , [2] J. Lunze and J. Richter, Reconigurable ault-tolerant control: A tutorial introduction, European Journal o Control, vol. 5, pp , [3] M. Verhaegen, S. Kanev, R. Hallouzi, C. Jones, J. Maciejowski, and H. Smaili, Fault tolerant light control - A survey, in Fault Tolerant Flight Control, ser. Lecture Notes in Control and Inormation Sciences, C. Edwards, T. Lombaerts, and H. Smaili, Eds. Springer, 2010, vol. 399, ch. 2, pp [4] Z. Gao and P. 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