Nonlinear observer techniques for oscillatory failure case detection

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  • NONLINEAR OBSERVER TECHNIQUESFOR OSCILLATORY FAILURE CASE DETECTION

    D. Emov1 and A. Zolghadri2

    1INRIA LNEParc Scientique de la Haute Borne

    40 Avenue Halley, Bat. A Park Plaza, Villeneuve dAscq 59650, France2University of Bordeaux, IMS-lab, Automatic Control Group

    351 Cours de la Liberation, Talence 33405, France

    The problem of observer design for fault detection in a class of nonlin-ear systems subject to parametric and signal uncertainties is studied.The design procedure includes formalized optimization of observer freeparameters in terms of trade-os for fault detection performance and ro-bustness to external disturbances or model uncertainties. The techniquemakes use of some monotonicity conditions imposed on the estimationerror dynamics. Eciency of the proposed approach is demonstratedthrough the Oscillatory Failure Case (OFC) in aircraft control surfaceservoloops.

    1 INTRODUCTION

    Model-based Fault Detection and Isolation (FDI) in dynamical systems is anactive research area (for a recent survey, see [1,2]). An important focus has beenmade on the use of observer-based schemes. In the linear case, it has been shownthat any linear fault detection lter can be transformed into an observer-basedform [3], providing a unied framework for analysis and implementation [48].From an estimation point of view, the problem of optimal noise ltering forstochastic linear systems has many solutions [9, 10]. For nonlinear systems, ageneral framework does not exist, although numerical or suboptimal solutionsare available [1113]. Typically, the observer design problem is solvable for acanonical representation of nonlinear systems [14,15].In this paper, an approach is developed for nonlinear fault detection observer

    design, together with a procedure for parameter tuning. For the latter, the designis made under monotonicity assumption [16] for the estimation error dynamic.

    Progress in Flight Dynamics, GNC, and Avionics 6 (2013) 375-392 DOI: 10.1051/eucass/201306375 Owned by the authors, published by EDP Sciences, 2013

    This is an Open Access article distributed under the terms of the Creative Commons Attribution License 2.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

    Article available at http://www.eucass-proceedings.eu or http://dx.doi.org/10.1051/eucass/201306375

    http://www.eucass-proceedings.euhttp://dx.doi.org/10.1051/eucass/201306375

  • PROGRESS IN FLIGHT DYNAMICS, GNC, AND AVIONICS

    In this case, using an appropriate linear parameter varying (LPV) transforma-tion [1719], the design of minorant and majorant monotone linear systems ispossible, whose solutions create an envelope for the original system trajectories.Solving an optimization problem for the minorant and majorant systems (thesolution is straightforward due to their linearity), it is possible to obtain a sub-optimal solution for the original nonlinear system (a local optimality is ensuredwhen the system solutions converge to the minorant or majorant ows). Thegoal is to maximize robustness with respect to disturbances and sensitivity withrespect to faults.In some cases, the faulty signal is known to belong to a specic class of

    signals (e. g., the harmonic functions of time with predened frequency range).For example, in the ODC detection considered in [20], the faults are assumed tobe harmonic. Such a priori available information simplies searched solution,since specied techniques oriented on analysis of the harmonic input responsecan be used for design of observer gains.The following nonlinear system is considered:

    x = Ax+GF(Hx,u+ f , ) + Sv ; y = Cx+ d (1)

    where x Rn, u Rm, and y Rp are the system state, input, and output;v Rv and d Rp are the state and the output disturbances; f Rm is thefaulty signal (unknown portion of the input); Rq is the vector of unknownparameters; the matrices A, G, H, S, and C are known and constant havingappropriate dimensions; and the function F : Rl+m+q Rg is continuouslydierentiable. The matrices G and H are introduced to take into account moreaccurately the inuence of nonlinearity on the system behavior. For simplicityof presentation, the signal f is considered to act on the control signal as additivedisturbance. Such a restriction is motivated by the numerical example studyfrom aeronautic eld in section 6. However, the approach can be applied tomultiplicative faults also. Assume that all input signals u, f, v, and d are(Lebesgue) measurable and essentially bounded, i. e.,

    f ess supt0|f(t)| < + .

    The objective is to design an observer for (1) using the available noisy mea-surements y and the input u, and ensuring robustness with respect to the un-certain parameters and the signals v and d. Moreover, for fault detection, it isrequired to nd the observer gains maximizing sensitivity of the output estima-tion error with respect to f and robustness with respect to v and d. Note thatif the fault detection problem is not of interest, then f can be considered as anadditional unknown input.Two solutions of this problem are presented below. One is more conventional

    and it is based on LMIs verication (see section 4). It is shown that optimization

    376

  • FAULT DETECTION AND CONTROL

    in this framework is complicated. Another solution is the main contribution ofthe work, and it utilizes the monotone system routine for analysis and optimiza-tion (see section 5).The paper is organized as follows. Preliminaries are given in section 2. The

    observer equations are introduced in section 3. Stability conditions based onLMIs are presented in section 4 (the optimization possibilities of this approachare also discussed). An alternative approach (monotone system theory) for theobserver stability analysis and the new optimization technique are given in sec-tion 5. In section 6, the overall approach is illustrated through its application toOFC detection in aircraft control surface servoloops.

    2 PRELIMINARIES

    This section introduces some basic notions about monotone systems and LPVrepresentation of nonlinearities.

    2.1 Linear Parameter Varying Representation of NonlinearFunctions

    For any two vectors p and p of the same dimension, let dene

    L(p,p) = {p+ (1 )p , 0 1}(the line connecting the points p and p). Since the vector valued function Fin (1) is continuously dierentiable, then according to the Mean Value Theorem,for any h,h HX , u,u U F , , , there exist hj L(h,h), uj L(u,u), j L(,), j = 1, g such that

    F(h,u,) F(h,u,) = x(h h) + u(u u) + ( ) ;

    x,j =Fj(,v,q)

    =xj ,v=

    uj ,q=

    j

    ; u,j =Fj(,v,q)

    v

    =xj ,v=

    uj ,q=

    j

    ;

    ,j =F(,v,q)

    q

    =xj ,v=

    uj ,q=

    j

    , j = 1, g

    where the symbols x,j, u,j , ,j denote the jth row of the correspondingmatrix.The application of this technique gives an exact equivalent LPV represen-

    tation of a nonlinear function. It is not a linearization around a single point(or around a trajectory) since the above expression is an equality. The LPV

    377

  • PROGRESS IN FLIGHT DYNAMICS, GNC, AND AVIONICS

    approach allows to transform nonlinear models to the linear ones depending onunknown parameters x, u, and . Therefore, the complexity of the nonlin-ear model (1) can be replaced with enlarged parametric uncertainty of a linearone. This tool will be applied in the next section to analyze the estimation errordynamics of the observer.

    2.2 Monotone System Theory

    The systemx = f(t,x) , x X , t 0 ,

    with the solution x(t,x0) for the initial condition x(0) = x0 is called monotone,if x0 0 x(t,x0) x(t, 0) for all t 0 [16] (for the vectors x0 and 0, theinequality x0 0 is understood elementwise). The system is called cooperativeif fi(t,x)/xj 0 for all 1 i = j n, t R and x X [16]. Cooperativesystems form a subclass of monotone ones. A matrix A with dimension n nis called Metzler if Ai,j 0 for all 1 i = j n. Note that for the cooperativestable system (the matrix A is Metzler and Hurwitz),

    s(t) = As(t) + r(t) , s Rn , r Rn , t 0 ,the properties s(0) 0, r(t) 0 for all t 0 imply s(t) 0 for t 0 and,conversely, s(0) 0, r(t) 0 for all t 0 ensures s(t) 0 for t 0. The systemis called competitive if fi(t,x)/xj 0 for all 1 i = j n, t R and x X ;in backward time, the competitive systems behave like the cooperative ones [16].

    3 ROBUST OBSERVER EQUATIONS

    This section is based on the following assumption.

    Assumption 1. Let the compact sets X Rn, U Rm, F Rm, V Rv,D Rp, and Rq be given such that for almost all t 0,

    x(t) X ; u(t) U ; f(t) F ; v(t) V ; d(t) D ; .

    Such constraints are rather common in nonlinear observer design theory stat-ing that the system (1) has bounded inputs and the state with some known upperbounds.Consider the following Luenberger type observer for (1):

    z = Z(z,y,u) = Az

    +GF[Hz+ L2(y Cz),u+ L3(y Cz), + L4(y Cz)] + L1(y Cz) (2)

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  • FAULT DETECTION AND CONTROL

    where z Rn is the estimate of the state x; Li, i = 1,4, are the observer gains tobe designed; and is a supporting xed value for the vector of unknownparameters. In (2), the output injection term is introduced for all arguments ofthe nonlinear function F. The gain L1 is standard, it is used to ensure stability ofthe pure linear part of the estimation error e = xz dynamics. The gain L2 hasbeen proposed in [21] in order to improve the robustness abilities of (2) and torelax restrictiveness of the LMIs used for the observer design. The gains L3 andL4 have been introduced in [22] to improve robustness of the system with respectto v, d, and sensitivity with respect to f . These gains have to be assigned toguarantee (or to nd a trade-o) the sy