State estimation and decoupling of unknown inputs in uncertain LPV systems using interval observers

  UPCommons Portal del coneixement obert de la UPC http://upcommons.upc.edu/e-prints © 2018 Taylor & Francis Group . Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works Aquesta és una còpia de la versió author’s final draft d'un article publicat a la revista International Journal of Control. URL d'aquest document a UPCommons E-prints: http://hdl.handle.net/2117/114115 Article publicat / Published paper : Rotondo, D., Cristofaro, A., Johansen , T. A., Nejjari, F., Puig, V. State estimation and decoupling of unknown inputs in uncertain LPV systems using interval observers. "International journal of control", 13 Juny 2017, p. 1-18. DOIE: 10.1080/00207179.2017.1335882 January 18, 2018 International Journal of Control IJC˙proportionalintervalUIO˙R2 To appear in the International Journal of Control Vol. 00, No. 00, Month 20XX, 1–26 State estimation and decoupling of unknown inputs in uncertain LPV systems using interval observers D. Rotondoa,b,∗, A. Cristofaro a,c, T. A. Johansen a, F. Nejjari band V. Puig b,d aCentre for Autonomous Marine Operations and Systems (AMOS), Department of Engineering Cybernetics, Norwegian University of Science and Technology, Trondheim, Norway; bResearch Center for Supervision, Safety and Automatic Control (CS2AC), Universitat Polit` ecnica de Catalunya (UPC), Rambla de Sant Nebridi, 22, Terrassa, Spain; cScuola di Scienze e Tecnologie, Universit` a di Camerino, Camerino (MC), Italy; dInstitut de Robotica i Informatica Industrial (IRI), UPC-CSIC, Carrer de Llorens i Artigas, 4-6, Barcelona, Spain. (Received 00 Month 20XX; accepted 00 Month 20XX) This paper proposes a linear parameter varying (LPV) interval observer for state estimation and unknown inputs decoupling in uncertain continuous-time LPV systems. Two different problems are considered and solved: i) the evaluation of the set of admissible values for the state at each instant of time; and ii) the unknown input observation, i.e. the design of the observer in such a way that some information about the nature of the unknown inputs affecting the system can be obtained. In both cases, analysis and design conditions, which rely on solving linear matrix inequalities (LMIs), are provided. The effectiveness and appeal of the proposed method is demonstrated using an illustrative application to a two-joint planar robotic manipulator. Keywords: Unknown input observers, linear parameter varying (LPV) systems, fault detection and isolation, uncertain systems, interval observers. 1. Introduction The problem of state estimation has been widely studied in the literature for both linear and nonlinear systems (Besanc¸on, 2007; Fossen and Nijmeijer, 1999; Meurer, Graichen, and Gilles, 2005). For example, an estimation of the state may be needed for control design or fault detection. When only the initial condition is assumed to be unknown, classical observers (Andrieu, Praly, and Astolfi, 2009; Luenberger, 1964) provide an estimation which converges asymptotically to the state of the considered system. However, the presence of uncertainties coming from either external disturbances or from the mismatch between the model and the real system may impede the convergence of classical state observers to the exact value of the state (Chebotarev, Efimov, Raissi, and Zolghadri, 2013; Efimov, Raissi, Perruquetti, and Zolghadri, 2013; Wang, Bevly, and Rajamani, 2015). In this situation, interval observers can be an appealing alternative approach (Gouz´ e, Rapaport, and Hadj-Sadok, 2000) because, under some assumptions, they can provide the set of admissible values for the state at each instant of time. Unlike stochastic approaches, such as the Kalman filter (Simon, 2006), interval observers ignore any probability distribution of the sources of uncertainty, and assume that they are constrained in a known bounded set. Using this information, instead of a single trajectory for each state variable, the interval observer computes the lower and upper bounds, which are compatible with the uncertainty (Raka and Combastel, 2013). There are several approaches for designing interval observers, e.g. the ones proposed by Jaulin (2002) and Kieffer and Walter (2004). A successful ∗Corresponding author. Email: [email protected] January 18, 2018 International Journal of Control IJC˙proportionalintervalUIO˙R2 framework for interval observer design is based on the monotone system theory, and has been proposed at first by Olivier and Gouz` e (2004), and further investigated by Moisan, Bernard, and Gouz` e (2009) and Ra¨ ıssi, Videan, and Zolghadri (2010); Ra¨ ıssi, Efimov, and Zolghadri (2012). Fault detection and diagnosis (FDD) is an important subfield of control engineering that aims at monitoring a system with the goal of identifying the occurrence of a fault, as well as to provide useful informations about the fault, e.g. its location (Ding, 2013; Witczak, 2014). A well-established FDD paradigm is the residual-based one, where one or more signals are created based on a model of the system and the knowledge of its inputs and outputs (Gao, Cecati, and Ding, 2015). Then, the analysis of this signal can help to determine which fault has occurred (Gertler, 1998). The residual-based FDD paradigm has been investigated thoroughly in the last decades, and several results are available, for both linear (Henry and Zolghadri, 2005) and nonlinear (Kabor´ e and Wang, 2001; Kabor´ e, Othman, McKenna, and Hammouri, 2000) systems. However, in order to increase the reliability and performance of this paradigm, robustness issues must be addressed, i.e. the fault diagnoser must only be sensitive to faults, even in the presence of model-reality mismatch (Chen and Patton, 1999). In this sense, the interval observer theory provides a passive approach for the development of a robust fault diagnoser, since the absence of false alarms and wrong diagnosis due to uncertainty and other undesired effects, e.g. noise, can be guaranteed by the property of interval estimation. Among the most successful techniques available in the literature for residual generation, there is the unknown input observer (UIO) approach (Kudva, Viswanadham, and Ramakrishna, 1980). UIOs are observers that allow estimating the state of a given system, independently of some unknown inputs (Hammouri and Tmar, 2010). One important feature of this approach is that UIOs can be made insensitive to certain input space directions if some structural conditions on the system are fulfilled (Cristofaro and Johansen, 2014). In this way, the decoupling between the external disturbances (unknown inputs) acting on the system and the estimation error can be attained, which is a very useful property that can be exploited for the purpose of FDD (Chen, Patton, and Zhang, 1996). Recent research has considered UIO design for FDD in nonlinear systems (Amato, Cosentino, Mattei, and Paviglianiti, 2006). The UIO proposed by Amato et al. (2006) has two relevant and appealing features: (i) the observer structure is nonlinear; and (ii) the effect of neglected nonlinearities, which for instance may represent structured uncertainty, and decoupled disturbances is minimized using a H∞optimization. However, the UIO in Amato et al. (2006) has the following shortcomings: (i) the proof of stability of the estimation error dynamics passes through the linearisation of the nonlinear term. Hence, it is theoretically valid only if the system state is in the neighbourhood of the estimated state; and (ii) a large nonlinear campaign of simulations with different kinds of faults and operating conditions is needed in order to obtain an appropriate tuning of suitable isolation thresholds. In contrast with linearisation techniques, linear parameter varying (LPV) methods have the advantage of not involving any approximation, since they can rely on an exact transformation of the original nonlinear system into a quasi-linear one, by embedding all the original nonlinearities within some varying parameters that schedule the state space matrices (Shamma, 2012). The LPV paradigm, which has attracted a lot of attention in the last decades (Hoffmann and Werner, 2014), provides an elegant way of guaranteeing theoretical stability and performance in nonlinear systems using linear-like techniques (Shamma and Athans, 1991). Hence, it is an appealing paradigm for the design of UIOs for nonlinear systems for which theoretical properties hold even in the presence of a mismatch between the system’s and the estimated state. At the same time, the interval observer paradigm is appealing because the estimated lower and upper bounds for the state can be used for generating unknown input isolation signals that embed the information about the uncertainty in such a way that a demanding simulation-based tuning of the isolation thresholds can be avoided. Motivated by the above mentioned properties, the goal of this work is to merge the theory of interval observers with the theory of unknown input observers, developing an interval UIO which can be applied to the problem of fault detection and isolation in uncertain LPV systems subject to faults and other undesired effects. Achieving this goal requires further modification of the solution proposed by Chebotarev et al. (2013) and Efimov et al. (2013). 2 January 18, 2018 International Journal of Control IJC˙proportionalintervalUIO˙R2 The paper is structured as follows. Section 2 introduces the two problems, that are solved in the subsequent sections. Problem 1, which is solved in Section 3, refers to the design of an LPV interval observer, which computes lower and upper bounds for the state, provided that no unknown inputs act on the observed system. On the other hand, Problem 2, which is solved in Section 4, deals with the presence of unknown inputs through the design of an LPV interval unknown input observer. Finally, Section 5 illustrates the application of the proposed approach and Section 6 presents the main conclusions. Notation: The set of (non-negative) real numbers will be denoted by R(R+). For a given vector signal u:R7→ Rnu, the shorthand notation utwill be used instead of u(t). Also, Lnu ∞will denote the set of all signals usuch that kuk∞=sup{|ut|,t∈R+}<∞. Given a matrix M∈Rm×n,He{M}will be used as a shorthand notation for M+MT. For two vectors x1,x2∈Rnor matrices M1,M2∈Rm×n, the relations x1≤x2and M1≤M2should be understood element-wise. The notation M†denotes the Moore-Penrose pseudo-inverse of M∈Rm×n. If M∈Rn×nis symmetric, then M∈Sn×n. The notation M≺0 (M0) means that M∈Sn×nis negative (positive) definite. If M∈Sn×nis diagonal, then M∈Dn×n. If all the elements of M∈Rn×noutside the main diagonal are non-negative, then M∈Mn×n(Metzler). For a generic vector x∈Rn, its i-th element will be denoted by x(i). For a given M∈Rm×nand a set of column indices N, with Na subset of {1,...,n}, the i-th column of Mwill be denoted by M(i), while M(N)will denote the matrix obtained from Mby replacing all columns whose indices do not belong to Nwith zeros. Also, the notation Π(M)xwill denote the projection of xonto the subspace generated by the columns of M. Given a set S, the notation P(S)will denote the power set of S, i.e. the set of all subsets of S, including the empty set and Sitself. Finally, given M∈Rm×n,M+=max{0,M}, where max denotes the element-wise maximum, M−=M+−M, and |M|=M++M−. 2. Problem statement Consider an uncertain LPV system described by: ˙xt=[A(ϑt)+∆A(ϑt)]xt+[B(ϑt)+ ∆B(ϑt)]ut+[Bun (ϑt)+∆Bun (ϑt)]uun,t+ct+dt(1) yt=Cxt(2) where x∈Rnxis the state, u∈Rnuis the known input (e.g. the control action), uun ∈Rnuun is the unknown input (e.g. some actuator fault), c∈Rnxis a known term, d∈Rnxis an unknown and unstructured disturbance and y∈Rnyis the output available from the sensors. The elements of the matrix functions appearing in (1) are nonlinear functions of some known time varying parameters, which are represented by the vector ϑt∈Θ⊂Rnϑ, where Θis a known closed and bounded set. Also, it is assumed that the derivatives of the scheduling parameters ˙ ϑtare known. It is assumed that the matrix C∈Rny×nxis full row rank, that the matrix functions A(ϑt),B(ϑt),Bun (ϑt)(of appropriate dimensions) are known, with Bun (ϑt)full column rank and rank (C(Bun(ϑt) +∆Bun(ϑt))) = nun ≤ny∀ϑt∈Θ, whereas ∆A(ϑt),∆B(ϑt)and ∆Bun (ϑt)are unknown and represent the modelling uncertainty. Notice that given a nonlinear state equation of the following type: ˙xt=f(xt,ut,uun,t)+ ct+dt(3) where fdepends on some uncertain parameters, it is possible to apply systematic approaches for the generation of equivalent LPV representations, e.g. the one described in Kwiatkowski, Boll, and Werner (2006), to both the state equation without uncertainty (i.e. using nominal values for the uncertain parameters) and the state equation with uncertainty, obtaining respectively: ˙xt=A(ϑt)xt+B(ϑt)ut+Bun(ϑt)uun,t+ct+dt(4) 3 January 18, 2018 International Journal of Control IJC˙proportionalintervalUIO˙R2 and: ˙xt=˜ A(ϑt)xt+˜ B(ϑt)ut+˜ Bun(ϑt)uun,t+ct+dt(5) Then, an equivalent uncertain LPV representation of (3) can be easily obtained by considering: ∆A(ϑt) = ˜ A(ϑt)−A(ϑt)(6) ∆B(ϑt) = ˜ B(ϑt)−B(ϑt)(7) ∆Bun(ϑt) = ˜ Bun(ϑt)−Bun(ϑt)(8) Without loss of generality, and up to a change of coordinates, it is possible to consider that Chas the following structure: C=˜ C0(9) where ˜ C∈Rny×nyis invertible. As recalled in the introduction, interval observers evaluate the set of admissible values for the state at each instant of time. In other words, an interval observer will provide two signals, namely the lower and the upper estimated bounds for the state, rather than a single one (the estimated state). Problem 1 concerns the extension of this concept to a structure for the interval observer which is suitable for the unknown input observation. Before stating the problem, let us introduce an assumption about the boundedness of disturbances and uncertainties, that will be required for establishing a solution. Assumption 1. There exist dt,dt∈Lnx ∞,∆A(ϑt),∆A(ϑt)∈Rnx×nxand ∆B(ϑt),∆B(ϑt)∈Rnx×nusuch that for all ϑt∈Θ: dt≤dt≤dt(10) ∆A(ϑt)≤∆A(ϑt)≤∆A(ϑt)(11) ∆B(ϑt)≤∆B(ϑt)≤∆B(ϑt)(12) Notice that since Θis closed and bounded, given a continuous matrix function R(ϑt)∈Rnx×nx, (10)-(12) are equivalent to the existence of dR,t,dR,t∈Lnx ∞,∆AR(ϑt),∆AR(ϑt)∈Rnx×nxand ∆BR(ϑt),∆BR(ϑt)∈Rnx×nu such that for all ϑt∈Θ: dR,t≤R(ϑt)dt≤dR,t(13) ∆AR(ϑt)≤∆AR(ϑt) = R(ϑt)∆A(ϑt)≤∆AR(ϑt)(14) ∆BR(ϑt)≤∆BR(ϑt) = R(ϑt)∆B(ϑt)≤∆BR(ϑt)(15) Problem 1. Given a continuous matrix function R(ϑt)∈Rnx×nx, partitioned as follows: R(ϑt) = R11(ϑt)0 0I(16) with R11(ϑt)∈Rny×nyand such that: AR12(ϑt)≥0 (17) AR22(ϑt)∈M(nx−ny)×(nx−ny)(18) 4 January 18, 2018 International Journal of Control IJC˙proportionalintervalUIO˙R2 where AR12(ϑt)∈Rny×(nx−ny)and AR22 denote the upper-right and lower-right sub-matrices of AR(ϑt) = R(ϑt)A(ϑt), respectively, determine an LPV interval observer which computes xtand ¯xtsuch that: xt≤xt≤¯xt∀t≥0 (19) with xt, ¯xt∈Lnx ∞, provided that: x0≤x0≤¯x0(20) uun,t=0∀t≥0 (21) and Assumption 1 holds. The parameter varying matrix function R(ϑt)is relevant to solve the problem of unknown input observation, which is formalized in Problem 2. In this case, in addition to solve Problem 1, the interval observer will also exhibit some desired properties of decoupling between the effects of the unknown inputs uun affecting the system. In this way, by looking at the projections of appropriate signals onto some subspaces, which are generated by the columns of an appropriate matrix H, it will be possible to detect the presence of unknown inputs acting on the system, as well as to identify their nature (isolation). In order to solve Problem 2, two additional assumptions are needed. Assumption 2 concerns the boundedness of signals and uncertainties related to the unknown inputs. On the other hand, Assumption 3 refers to the structure of the uncertainty that affects the state matrix, requiring that the non-measured states influence the measured ones in a known manner. Assumption 2. The signal uun,tis such that: uun,t≤uun,t≤¯uun,t(22) with uun,t≤0 and ¯uun,t≥0, uun,¯uun ∈Lnu ∞. Moreover, there exist ∆Bun(ϑt),∆Bun(ϑt)∈Rnx×nuun such that for all ϑt∈Θ: ∆Bun(ϑt)≤∆Bun(ϑt)≤∆Bun(ϑt)(23) Also in this case, since Θis closed and bounded, given a continuous matrix function R(ϑt)∈Rnx×nx, (23) is equivalent to the existence of ∆Bun,R(ϑt),∆Bun,R(ϑt)∈Rnx×nuun such that for all ϑt∈Θ: ∆Bun,R(ϑt)≤R(ϑt)∆Bun(ϑt)≤∆Bun,R(ϑt)(24) Assumption 3. The matrix ∆A(ϑt)is partitioned as: ∆A(ϑt) = ∆A11(ϑt)0 ∆A21(ϑt)∆A22(ϑt)(25) with ∆A11(ϑt)∈Rny×ny. Problem 2. Given an invertible matrix function R(ϑt)∈Rnx×nxpartitioned as in (16) and such that (17)- (18) hold, and a matrix H∈Rnx×nuun for which the following holds: R(ϑt)Bun(ϑt) = H∀ϑt∈Θ(26) and provided that (20) and Assumptions 1-3 hold, determine an LPV interval unknown input observer 5 January 18, 2018 International Journal of Control IJC˙proportionalintervalUIO˙R2 which, in addition to solve Problem 1, satisfies: u(j) un,t=0⇒Π(H(j))εt≥0∧Π(H(j))¯ εt≥0 (27) Π(H(j))εt<0∨Π(H(j))¯ εt<0⇒u(j) un,t,0 (28) where εtand ¯ εtare evaluable quantities that can be used as unknown input isolation signals. In particular, in this paper, it is shown that a valid choice for these signals is the following: εt=C†(yt−Cxt)(29) ¯ εt=C†(C¯xt−yt)(30) In other words, if the j-th unknown input has value equal to zero, the projections of both εtand ¯ εtonto the subspace generated by the j-th column of Hwill be non-negative. On the other hand, if at least one of such projections is negative, it means that the j-th unknown input has value different from zero, which allows performing a correct isolation. Remark 1: In presence of an uncertain term εt, the application of nonlinear fault diagnosis strategies as the one described in Kabor´ eet al. (2000) requires the knowledge of a uniform bound µ>0 such that ∀t:kεtk ≤ µ, with µknown a priori in order to calculate appropriate thresholds for the residuals. However, if the uncertainty is structured, as in the case detailed in this paper for which εt=∆A(ϑt)xt+∆B(ϑt)ut+ ∆Bun(ϑt)uun,t+dt, a description of the uncertainty as in Kabor´ eet al. (2000) can be overly conservative. This conservativeness is avoided by the interval-based approach detailed in the following, which exploits the structuredness of the uncertainty and uses elementwise bounds on the individual terms ∆A(ϑt),∆B(ϑt), ∆Bun(ϑt),uun,tand dt(in the case of quasi-LPV systems (Shamma and Athans, 1991) knowledge of bounds on the state xtis also needed for computing the set Θ). 3. LPV interval observer design 3.1 The LPV interval observer The LPV interval observer proposed to solve Problem 1 can be conveniently decomposed into two coupled subsystems, i.e. a lower bound observer, which provides xt, as follows: ˙zt=F(ϑt)zt+R(ϑt)B(ϑt)ut+S(ϑt)yt−˙ Tϑt,˙ ϑtyt+dR,t+ct−T(ϑt)Cct(31) +∆AR(ϑt)+x+ t−∆AR(ϑt)+x− t−∆AR(ϑt)−¯x+ t+∆AR(ϑt)−¯x− t +∆BR(ϑt)+u+ t−∆BR(ϑt)+u− t−∆BR(ϑt)−u+ t+∆BR(ϑt)−u− t xt=zt+T(ϑt)yt(32) and an upper bound observer, which provides ¯xt, as follows: ˙ ¯zt=F(ϑt)¯zt+R(ϑt)B(ϑt)ut+S(ϑt)yt−˙ Tϑt,˙ ϑtyt+dR,t+ct−T(ϑt)Cct(33) +∆AR(ϑt)+¯x+ t−∆AR(ϑt)+¯x− t−∆AR(ϑt)−x+ t+∆AR(ϑt)−x− t +∆BR(ϑt)+u+ t−∆BR(ϑt)+u− t−∆BR(ϑt)−u+ t+∆BR(ϑt)−u− t ¯xt=¯zt+T(ϑt)yt(34) where F(ϑt),F(ϑt),S(ϑt),S(ϑt)and T(ϑt)are matrix functions of appropriate dimensions, and ˙ T(ϑt,˙ ϑt) is obtained from T(ϑt)by differentiating each element with respect to time. 6 January 18, 2018 International Journal of Control IJC˙proportionalintervalUIO˙R2 The following theorem provides the conditions which should be met to ensure an interval estimation of xtand the boundedness of xt, ¯xt, as specified in Problem 1. Theorem 1: Let Assumption 1 be satisfied, x ∈Lnx ∞, u ∈Lnu ∞, c ∈Lnx ∞, the interval observer be given by (31)-(34), the matrix functions R(ϑt)∈Rnx×nxand F (ϑt), F (ϑt)∈Mnx×nxbe chosen such that R(ϑt)is partitioned as in (16),(17)-(18) hold, and: F12(ϑt) F22(ϑt)=F12(ϑt) F22(ϑt)=AR12(ϑt) AR22(ϑt)(35) where F12(ϑt),F12(ϑt)∈Rny×(nx−ny)and F22(ϑt),F22(ϑt)∈M(nx−ny)×(nx−ny)denote the upper-right and lower-right sub-matrices of F(ϑt)and F(ϑt), respectively. Then, the relation (19) is satisfied provided that (20)-(21) hold and the matrix functions T(ϑt),S(ϑt),S(ϑt)∈Rnx×nyare chosen as1: T(ϑt)C=I−R(ϑt)(36) S(ϑt) = S1(ϑt)+S2(ϑt)(37) S(ϑt) = S1(ϑt)+S2(ϑt)(38) S1(ϑt)C=R(ϑt)A(ϑt)−F(ϑt)(39) S1(ϑt)C=R(ϑt)A(ϑt)−F(ϑt)(40) S2(ϑt) = F(ϑt)T(ϑt)(41) S2(ϑt) = F(ϑt)T(ϑt)(42) In addition, if there exist P,Q∈S2nx×2nx, P,Q0and constants ε1,ε2,γ>0such that the following matrix inequality is verified: Φ(ϑt) = G(ϑt)TP+PG(ϑt)+(ε1+ε2)P+Q+γη(ϑt)2I2nx0 0ε−1 1P−γI2nx0 (43) where: η(ϑt) = 2  ∆AR(ϑt)+−∆AR(ϑt)+  2+  ∆AR(ϑt)−  2+  ∆AR(ϑt)−  2(44) G(ϑt) = F(ϑt)+∆AR(ϑt)+0 0F(ϑt) +∆AR(ϑt)+!(45) then xt, xt∈Lnx ∞. The theorem statement consists of two parts. Eqs. (36)-(42) guarantee that, at each instant of time, the true state of the LPV system (1)-(2) will lie inside the region defined by the lower and upper estimates. On the other hand, the feasibility of the matrix inequality (43) ensures that such estimates will remain bounded, i.e. they will not diverge. Proof of Theorem 1: Let us consider the dynamics of the interval estimation errors et=xt−xtand 1Notice that the existence of matrix functions T(ϑt),S(ϑt),S(ϑt)satisfying (36)-(42) is guaranteed by the fact that (16)-(18) and (35) hold. 7 January 18, 2018 International Journal of Control IJC˙proportionalintervalUIO˙R2 ¯et=¯xt−xtwhich, taking into account (1)-(2), (31)-(34) and (36)-(42), become: ˙et=F(ϑt)et+R(ϑt)(Bun (ϑt) +∆Bun (ϑt))uun,t+ 3 ∑ i=1 wi t(46) ˙ ¯et=¯ F(ϑt)¯et−R(ϑt)(Bun (ϑt)+ ∆Bun (ϑt))uun,t+ 3 ∑ i=1 ¯wi t(47) where: w1 t=R(ϑt)dt−dR,t(48) w2 t=∆AR(ϑt)xt−∆AR(ϑt)+x+ t+∆AR(ϑt)+x− t+∆AR(ϑt)−¯x+ t−∆AR(ϑt)−¯x− t(49) w3 t=∆BR(ϑt)ut−∆BR(ϑt)+u+ t+∆BR(ϑt)+u− t+∆BR(ϑt)−u+ t−∆BR(ϑt)−u− t(50) ¯w1 t=dR,t−R(ϑt)dt(51) ¯w2 t=∆AR(ϑt)+¯x+ t−∆AR(ϑt)+¯x− t−∆AR(ϑt)−x+ t+∆AR(ϑt)−x− t−∆AR(ϑt)xt(52) ¯w3 t=∆BR(ϑt)+u+ t−∆BR(ϑt)+u− t−∆BR(ϑt)−u+ t+∆BR(ϑt)−u− t−∆BR(ϑt)ut(53) When (21) holds, since F(ϑt),F(ϑt)∈Mnx×nx, then any solution of (46)-(47) is element-wise non-negative for all t≥0, i.e. (19), provided that e0≥0, ¯e0≥0, wi t≥0 and ¯wi t≥0∀t≥0, ∀i=1,2,3 (Farina and Rinaldi, 2000). e0≥0 and ¯e0≥0 hold due to (20). The terms w1 t, ¯w1 tare non-negative ∀t≥0 due to Assumption 1 (see (13)). On the other hand, w2 t, ¯w2 tremain non-negative as long as (19) holds, according to Lemma 1 in Efimov et al. (2013) and Assumption 1 (see (14)). (19) holds for t=0, due to e0≥0, ¯e0≥0, and (19) is preserved ∀t≥0 by induction, as long as w3 t, ¯w3 tremain non-negative too. Indeed, also w3 t, ¯w3 tremain non-negative because of Lemma 1 in Efimov et al. (2013) and Assumption 1 (see (15)). Let us show that the variables xtand ¯xtstay bounded ∀t≥0. For this purpose, let us notice that the equations that describe the dynamics of xtand ¯xtcan be rewritten as: ˙xt=F(ϑt) +∆AR(ϑt)+xt+f(xt,¯xt) +δt(xt,ut,uun,t,ct,dt)(54) ˙ ¯xt=F(ϑt)+∆AR(ϑt)+¯xt+f(xt,¯xt)+ ¯ δt(xt,ut,uun,t,ct,dt)(55) for some δt(·)and ¯ δt(·), with: f(xt,¯xt) = ∆AR(ϑt)+−∆AR(ϑt)+x− t−∆AR(ϑt)−¯x+ t+∆AR(ϑt)−¯x− t(56) f(xt,¯xt) = ∆AR(ϑt)+−∆AR(ϑt)+¯x− t−∆AR(ϑt)−x+ t+∆AR(ϑt)−x− t(57) Clearly, for all ϑt∈Θ,fand fsatisfy: f(xt,¯xt)≤  ∆AR(ϑt)+−∆AR(ϑt)+  2|xt|+  ∆AR(ϑt)−  2+  ∆AR(ϑt)−  2|¯xt|(58) f(xt,¯xt)≤  ∆AR(ϑt)+−∆AR(ϑt)+  2|¯xt|+  ∆AR(ϑt)−  2+  ∆AR(ϑt)−  2|xt|(59) and, if (21) holds, the inputs δt,δtare bounded due to Assumption 1 and the fact that x∈Lnx ∞,u∈Lnu ∞, c∈Lnx ∞. 8 January 18, 2018 International Journal of Control IJC˙proportionalintervalUIO˙R2 F(ϑt)H=HΓ(ϑt)(104) where Γ(ϑt),Γ(ϑt)∈Rnuun ×nuun contain some of the eigenvalues of F(ϑt),F(ϑt)(the ones that correspond to the eigenvectors that are columns of H). Taking into account (26), it is easy to see that (103)-(104) are equivalent to: F(ϑt)R(ϑt)Bun(ϑt) = HΓ(ϑt)(105) F(ϑt)R(ϑt)Bun(ϑt) = HΓ(ϑt)(106) Since Bun(ϑt)is full column rank, solutions to the matrix equations (105)-(106) exist. These solutions can be expressed as (101)-(102), which completes the proof.  The infinite number of conditions given by Theorem 2 can be brought to a finite number by gridding the varying parameter space Θusing Npoints ϑi,i=1,...,N, as already suggested in Section 3. 4.2 Design conditions Also in this case, it is possible to derive conditions for performing the design, as specified by the following corollary. Corollary 2: Given the matrix functions Γ(ϑt),Γ(ϑt)∈Dnuun ×nuun , let Assumptions 1-3 be satisfied, x∈Lnx ∞, u ∈Lnu ∞, c ∈Lnx ∞, the invertible matrix function R(ϑt)be partitioned as in (16) and such that (17)-(18) hold, and the matrix H ∈Rnx×nuun be such that (26) holds. Also, let us assume that there exist an element-wise non-negative block-diagonal matrix P as in (61), with P,P∈Snx×nx, P,P0, a matrix function: WH(ϑt) = WH(ϑt)0 0WH(ϑt)(107) with WH(ϑt),WH(ϑt)∈Rnx×nx, a matrix Q ∈S2nx×2nx, Q 0, a sufficiently large matrix function Σ∈ D2nx×2nx +and constants ε1,ε2,γ>0such that: P    AR12(ϑt)0 AR22(ϑt)0 0AR12(ϑt) 0AR22(ϑt)     =PF∗(ϑt)0 0F∗(ϑt)+WH(ϑt)    0 0 Inx−ny0 0 0 0Inx−ny     (108) and ∀S1,S2∈P({1,...,ny}): He{PΞ(ϑt,S1,S2)+WH(ϑt)ϒ(ϑt)}+ (ε1+ε2)P+Q+γη(ϑt,S1,S2)2I2nx0 0ε−1 1P−γI2nx0 (109) P  Ξ(ϑt,S2)−∆A(S2∪S3) R(ϑt)+0 0Ξ(ϑt,S1)−∆A(S1∪S3) R(ϑt) + +WH(ϑt)ϒ(ϑt)+PΣ(ϑt)≥0 (110) where Inx−nyis the identity matrix of order nx−ny, F∗(ϑt),F∗(ϑt)∈Rnx×(nx−ny)denote the right submatrices of R(ϑt)−1HΓ(ϑt)Bun(ϑt)†and R(ϑt)−1HΓ(ϑt)Bun(ϑt)†, respectively, η(ϑt,S1,S2)is defined 15 January 18, 2018 International Journal of Control IJC˙proportionalintervalUIO˙R2 as in (75) and: Ξ(ϑt,S1,S2) = Ξ(ϑt,S2)0 0Ξ(ϑt,S1)(111) ϒ(ϑt) = I−Bun(ϑt)Bun(ϑt)†0 0I−Bun(ϑt)Bun(ϑt)†(112) Ξ(ϑt,S2) = R(ϑ)−1HΓ(ϑt)Bun(ϑt)†+∆A(S2∪S3) R(ϑt)+(113) Ξ(ϑt,S1) = R(ϑt)−1HΓ(ϑt)Bun(ϑt)†+∆A(S1∪S3) R(ϑt) + (114) Then, the interval unknown input observer (31),(33) and (69)-(72), with F(ϑt)and F(ϑt)calculated as in (101)-(102), with: H∗(ϑt) =R(ϑt)P−1WH(ϑt)(115) H∗(ϑt) =R(ϑt)P−1WH(ϑt)(116) and T(ϑt),S(ϑt),S(ϑt)chosen as (36)-(42) is such that the relations (27)-(28) are satisfied provided that (20) holds. Moreover, if (21) holds, then also (19) is satisfied, with xt,xt∈Lnx ∞. Proof of Corollary 2: (109) can be obtained from (74) through the change of variables: WH(ϑt) = PR(ϑt)−1H∗(ϑt)0 0PR(ϑt)−1H∗(ϑt)(117) which explains why H∗(ϑt)and H∗(ϑt)are calculated as in (115)-(116). On the other hand, (108) and (110) corresponds to the verification of (35) and the Metzler property, respectively.  As already discussed previously, by gridding the varying parameter space Θusing Npoints ϑi,i= 1,...,N, (109) and (110) can be reduced to a finite set of LMIs, by requiring that they hold ∀ϑi,i=1,...,N. Remark: The proposed LPV interval unknown input observer follows the passive approach to robust fault diagnosis Chen and Patton (1999), which ensures that as long as the assumptions about bounds on uncertainties, disturbances and noise are satisfied, if no unknown inputs are acting on the system then the state will always be contained within the computed bounds (absence of false alarms). On the other hand, if some unknown inputs are acting on the system, only the corresponding components of the unknown input isolation signals might become negative (absence of wrong diagnosis). Anyway, as suggested by Ding (2013), it is possible to enhance the robustness against disturbances and the sensitiveness to faults by considering a multiobjective optimization. Commonly employed performance indices are the H∞norm and the H−index, which are minimized and maximized, respectively (Chadli, Abdo, and Ding, 2012; Henry, Cieslak, Zolghadri, and Efimov, 2015). However, considering a multiobjective H∞/H−optimization for the design of the LPV interval UIO goes beyond the scope of this paper, and will be addressed by future work. 4.3 Performance assessment In this section, a metric based on the idea of stochastic robustness (Marrison and Stengel, 1997; Witczak and Pretki, 2007) is proposed in order to assess the performance of the LPV interval UIO. This metric is given 16 January 18, 2018 International Journal of Control IJC˙proportionalintervalUIO˙R2 by the probability that the LPV interval UIO will exhibit an unacceptable behaviour. More specifically, let us denote the LPV interval UIO as O, while the set of possible scenarios is denoted by S(µ), where µ∈M denotes possible variations due to different realization of the model uncertainty, unknown inputs, etc. within a bounded set M, which can be described by a probability density function pr(µ). Then, the performance metric can be defined as the integral of an indicator function over the space of expected variations: Ψ(O) = ZM I[S(µ),O]pr(µ)dµ(118) where Iis a binary function which describes if the behaviour of the fault/icing diagnoser for a given realization of µis acceptable (I=1) or not (I=0). Unfortunately, (118) cannot be integrated analytically. A practical alternative is to use Monte Carlo methods (Doucet, de Freitas, and Gordon, 2001) with pr(µ)shaping random values of µthat will be denoted by µi. When Mrandom µi,i=1,...,Mare generated, then an estimate of Ψis given by: ˆ Ψ(O) = 1 M M ∑ i=1 I[S(µi),O](119) where ˆ Ψapproaches Ψin the limit as M→∞. However, it is impossible to set M=∞, thus it is interesting to choose Min such a way that ˆ Ψhas standard deviation less than a desired value σˆ Ψ. Since Iis binary, ˆ Ψ has a binomial distribution, such that Mcan be chosen as (Witczak and Pretki, 2007): M≥1 4σ−2 ˆ Ψ(120) 5. Application to a two-joint planar robotic manipulator Let us consider a two-joint planar robotic manipulator, for which the dynamics equation can be expressed as (Yu, Chen, and Woo, 2002): ˜a¨q1+˜ bcos(q2−q1)¨q2−˜ b˙q2 2sin(q2−q1) = τ1(121) ˜ bcos(q2−q1)¨q1+˜c¨q2+˜ b˙q2 1sin(q2−q1) = τ2(122) where q1and q2represent the positions of the first and the second joint, respectively, τ1and τ2are the joint torques, and ˜a,˜ b,˜care coefficients which depend on dynamic and kinematic parameters. It is assumed that ˜a=a+∆a,˜ b=b+∆b, ˜c=c+∆c, where a,b,care known (nominal coefficient values) and ∆a,∆b,∆c represent the uncertainty, which take unknown values in known intervals [∆a,∆a],[∆b,∆b]and [∆c,∆c], respectively. Similarly, τ1=τ∗ 1+∆τ1and τ2=τ∗ 2+∆τ2, where τ∗ 1,τ∗ 2are known inputs and ∆τ1,∆τ2are unknown (they can represent faults in the actuators). It is assumed that the full state is available for measurement, i.e. C=I. By using the state vector x= [q1,q2,˙q1,˙q2]T, the input vector u= [τ∗ 1,τ∗ 2]Tand the unknown input vector uun = [∆τ1,∆τ2]T, (121)-(122) can be brought to the form (5) with ϑt=xt,ct=0, dt=0, and: ˜ A(ϑt) =     0 0 1 0 0 0 0 1 0 0 ˜a33(ϑt)˜a34(ϑt) 0 0 ˜a43(ϑt)˜a44(ϑt)     ˜ B(ϑt) = ˜ Bun (ϑt) =     0 0 0 0 ˜ b31 (ϑt)˜ b32 (ϑt) ˜ b41 (ϑt)˜ b42 (ϑt)     17 January 18, 2018 International Journal of Control IJC˙proportionalintervalUIO˙R2 where the elements of the state and input matrices are given by: ˜a33(ϑt) = ˜ b2sin(x2−x1)cos(x2−x1)x3 ˜a˜c−˜ b2cos2(x2−x1) ˜a34(ϑt) = ˜ b˜csin(x2−x1)x4 ˜a˜c−˜ b2cos2(x2−x1) ˜a43(ϑt) = −˜a˜ bsin(x2−x1)x3 ˜a˜c−˜ b2cos2(x2−x1) ˜a44(ϑt) = −˜ b2sin(x2−x1)cos(x2−x1)x4 ˜a˜c−˜ b2cos2(x2−x1) ˜ b31(ϑt) = ˜c ˜a˜c−˜ b2cos2(x2−x1) ˜ b32(ϑt) = ˜ b41(ϑt) = −˜ bcos(x2−x1) ˜a˜c−˜ b2cos2(x2−x1) ˜ b42(ϑt) = ˜a ˜a˜c−˜ b2cos2(x2−x1) Then, as explained in Section 2, by neglecting the uncertainty, (4) is obtained with: A(ϑt) =     0 0 1 0 0 0 0 1 0 0 a33(ϑt)a34(ϑt) 0 0 a43(ϑt)a44(ϑt)     B(ϑt) = Bun (ϑt) =     0 0 0 0 b31 (ϑt)b32 (ϑt) b41 (ϑt)b42 (ϑt)     where the elements of A(ϑt),B(ϑt),Bun(ϑt)can be obtained from the corresponding elements of ˜ A(ϑt),˜ B(ϑt),˜ Bun(ϑt)by replacing ˜a,˜ b,˜cwith a,b,c, respectively. Then, ∆A(ϑt),∆B(ϑt),∆Bun(ϑt)can be obtained as in (6)-(8): ∆A(ϑt) =     0 0 0 0 0 0 0 0 0 0 ∆a33(ϑt)∆a34(ϑt) 0 0 ∆a43(ϑt)∆a44(ϑt)     ∆B(ϑt) = ∆Bun (ϑt) =     0 0 0 0 ∆b31 (ϑt)∆b32 (ϑt) ∆b41 (ϑt)∆b42 (ϑt)     For the sake of brevity, the expressions of the elements of ∆A(ϑt),∆B(ϑt),∆Bun(ϑt)are omitted, except for the illustrative example of ∆a33(ϑt), which is given by: ∆a33(ϑt) = (b+∆b)2sin(x2−x1)cos(x2−x1)x3 (a+∆a)(c+∆c)−(b+∆b)2cos2(x2−x1)−b2sin(x2−x1)cos(x2−x1)x3 ac −b2cos(x2−x1) 18 January 18, 2018 International Journal of Control IJC˙proportionalintervalUIO˙R2 The following step for the application of the proposed strategy is to find the lower and upper bounds such that (11)-(12) and (23) hold. These bounds can be found elementwise, taking into account the knowledge about the uncertainty intervals. For example, the element ∆a33(ϑt)can be bounded by: ∆a33(ϑt) =    (b+∆b)2cos(x2−x1)sin(x2−x1)x3 (a+∆a)(c+∆c)−(b+∆b)2cos2(x2−x1)−a33(ϑt)i f cos(x2−x1)sin(x2−x1)x3≥0 (b+∆b)2cos(x2−x1)sin(x2−x1)x3 (a+∆a)(c+∆c)−(b+∆b)2cos2(x2−x1)−a33(ϑt)i f cos(x2−x1)sin(x2−x1)x3<0 ∆a33(ϑt) =    (b+∆b)2cos(x2−x1)sin(x2−x1)x3 (a+∆a)(c+∆c)−(b+∆b)2cos2(x2−x1)−a33(ϑt)i f cos(x2−x1)sin(x2−x1)x3≥0 (b+∆b)2cos(x2−x1)sin(x2−x1)x3 (a+∆a)(c+∆c)−(b+∆b)2cos2(x2−x1)−a33(ϑt)i f cos(x2−x1)sin(x2−x1)x3<0 By choosing: R(ϑt) =     0 0 a bcos(x2−x1) 0 0 bcos(x2−x1)c 1 0 0 0 0 1 0 0     the matrix Hcalculated as in (26) is: H=    1 0 0 1 0 0 0 0     which means that if the first component of either εtor εtbecomes negative, u(1) un,t,0, while if the second component becomes negative, then u(2) un,t,0. It is easy to check that the choice of R(ϑt)leads to: ∆AR(ϑt) =     0 0 a∆a33(ϑt) +bcos(x2−x1)∆a43(ϑt)a∆a34(ϑt) +bcos(x2−x1)∆a44(ϑt) 0 0 bcos(x2−x1)∆a33(ϑt)+c∆a43(ϑt)bcos(x2−x1)∆a34(ϑt)+ c∆a44(ϑt) 0 0 0 0 0 0 0 0     ∆BR(ϑt) = ∆Bun,R(ϑt) =     a∆b31(ϑt)+bcos(x2−x1)∆b41(ϑt)a∆b32(ϑt) +bcos(x2−x1)∆b42(ϑt) bcos(x2−x1)∆b31(ϑt)+c∆b41(ϑt)bcos(x2−x1)∆b32(ϑt)+ c∆b42(ϑt) 0 0 0 0     Then, the bounds in (14)-(15) and (24) can be easily calculated. For example: ∆aR13(ϑt) = (a∆a33(ϑt)+bcos(x2−x1)∆a43(ϑt)i f cos(x2−x1)≥0 a∆a33(ϑt)+bcos(x2−x1)∆a43(ϑt)i f cos(x2−x1)<0 19 January 18, 2018 International Journal of Control IJC˙proportionalintervalUIO˙R2 Through the choices: Γ(ϑt) = Γ(ϑt) = Γ=−λI H∗(ϑt) = H∗(ϑt) = H∗=    0 0 0 0 0 0 0 0 −λ0 0 0 0−λ0 0     the matrices calculated using (101)-(102) are F(ϑt) = F(ϑt) = −λI. Notice that, according to (46)-(47), the matrices F(ϑt),F(ϑt), i.e. the choice of λ, will determine the dynamical behaviour of the interval estimation errors et,et. By using (36)-(42), the matrix functions T(ϑt),S(ϑt)and S(ϑt)can be calculated. For example: T(ϑt) =     1 0 −a−bcos(x2−x1) 0 1 −bcos(x2−x1)−c −1 0 1 0 0−1 0 1     which leads to: ˙ T(ϑt) =     0 0 0 b(x4−x3)sin(x2−x1) 0 0 b(x4−x3)sin(x2−x1)0 0 0 0 0 0 0 0 0     Then, according to the first part of Theorem 2, (31), (33) and (69)-(72) is an LPV interval unknown input observer for the considered system. Let us consider the following values: a=5.75kg ·m2,b=1.5kg ·m2,c=1.75kg ·m2,∆a=−∆a= 0.2kg·m2,∆b=−∆b=0.1kg·m2,∆c=−∆c=0.1kg·m2, ˜a=5.6kg·m2,˜ b=1.4kg·m2, ˜c=1.7kg·m2, λ=1 (notice that for any possible value of ˜a,˜ b, ˜c, ˜a˜c−˜ b2cos2(x2−x1)>0, such that the matrix functions ˜ A(ϑt),˜ B(ϑt)and ˜ Bun(ϑt)are well-defined). By gridding Θ= [−π,π]×[−π,π]×[−1,1]×[−1,1] into 10000 points, (74) can be assessed, thus confirming that xtand ¯xtwill stay bounded. For simulation purposes, let us consider: uun,t= [−5,−5]T, ¯uun,t= [5,5]T,x0= [π/12,−π/12,0,0]T,ξ0= [−π/6,−π/6,−π/12,−π/12]T,ξ0= [π/6,π/6,π/12,π/12]T. The control input utis provided by the LPV controller in Yu et al. (2002). Four different scenarios are considered: Scenario 1: uun,t= [0,0]T Scenario 2: uun,t=[0,0]Tt≤20s [5,0]Telse Scenario 3: uun,t=[0,0]Tt≤20s [0,5]Telse Scenario 4: uun,t=[0,0]Tt≤20s [5,5]Telse Figs. 1-4 show the responses of the unknown input isolation signals εtand ¯ εtin the four considered scenarios. As expected, in Scenario 1, all the components of εtand ¯ εtare positive, since no unknown input is acting on the system. On the other hand, in Scenario 2, ε(1) tbecomes negative at time t=21.16s, which allows detecting and isolating correctly the presence of the first unknown input. Similarly, in Scenario 3, ¯ ε(2) tbecomes negative at time t=20.02s, which means that the second unknown input is acting on the 20 January 18, 2018 International Journal of Control IJC˙proportionalintervalUIO˙R2 system. Finally, the change of sign of both ε(1) tand ¯ ε(2) tin Scenario 4 (see Fig. 4) confirms that both the unknown inputs are acting on the system at the same time, providing further confirmation of the validity of the developed theory. time (s) 0 5 10 15 20 25 30 35 40 45 50 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 ε(1) t ε(2) t ¯ε(1) t ¯ε(2) t Figure 1.: Unknown input isolation signals εtand εtin Scenario 1. time (s) 0 5 10 15 20 25 30 35 40 45 50 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 ε(1) t ε(2) t ¯ε(1) t ¯ε(2) t Figure 2.: Unknown input isolation signals εtand εtin Scenario 2. Finally, in order to assess the performance of the proposed method, the approach described in Section 4.3 has been applied, by performing Monte Carlo simulation with different values of the uncertainty and the unknown inputs. To this end, the metric (118) has been calculated by using an indicator function Ithat takes into account whether a given simulation has been successful or not. For example, in scenario 1, the simulation is considered to be successful if neither ε(i) tnor ε(i) t,i=1,2, becomes negative. On the other hand, in scenario 2, the success is characterized by ε(1) ior ε(1) tbecoming negative while both ε(2) iand ε(2) i remaining non-negative. For each considered scenario (scenarios 1-4, depending on which unknown inputs are affecting the system), uncertainty level (expressed as a percentage of the nominal parameters’ values) and unknown input magnitudes, M=100 Monte Carlo simulations have been performed which, according to (120), corresponds to ensuring a standard deviation σˆ Ψ=0.05. The results of the performance assessment are summarized in Figs. 5-8. Notably, in scenario 1 (Fig. 5), a performance metric ˆ Ψ(O) = 1 is obtained in all cases, since the proposed technique ensures the absence of false alarms. In all the other scenarios, it is evident that the higher is the uncertainty, the bigger is the value of the minimum detectable unknown input. 21 January 18, 2018 International Journal of Control IJC˙proportionalintervalUIO˙R2 time (s) 0 5 10 15 20 25 30 35 40 45 50 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 ε(1) t ε(2) t ¯ε(1) t ¯ε(2) t Figure 3.: Unknown input isolation signals εtand εtin Scenario 3. time (s) 0 5 10 15 20 25 30 35 40 45 50 -1 0 1 2 3 4 5 ε(1) t ε(2) t ¯ε(1) t ¯ε(2) t Figure 4.: Unknown input isolation signals εtand εtin Scenario 4. 6. Conclusions This paper has introduced the use of LPV interval observers for the state estimation in uncertain continuoustime LPV systems. The conditions for analysis and design of these observers are based on LMIs, which can be solved efficiently using available solvers. In particular, two properties are required by the analysis/design: i) interval estimation of the state, i.e. as long as some assumptions about uncertainties and disturbances are verified, the state will always be contained within the bounds calculated by the interval observer; and ii) boundedness of the estimation, which is akin to the asymptotic stability of classical state observers, and is verified by finding an appropriate Lyapunov function. Furthermore, it has been shown that a slight modification of the LPV interval observer allows decoupling unknown inputs acting on the system. In this way, an LPV interval unknown input observer is obtained. This unknown input observer is useful for the task of isolating faults and other undesired effects, because different output directions of the residuals can be assigned to these effects. Due to the property of interval estimation guaranteed by the observer, the absence of false alarms and wrong diagnosis will be assured. The application of the proposed approach to a two-joint planar robotic manipulator has demonstrated its appeal, giving more insight into this method and confirming the results provided by the theory. As previously remarked, future research will aim at considering a multiobjective H∞/H−optimization for the design of the LPV interval UIO with the aim of enhancing the robustness against disturbances 22 January 18, 2018 International Journal of Control IJC˙proportionalintervalUIO˙R2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Uncertainty % -0.2 0 0.2 0.4 0.6 0.8 1 1.2 ˆ Ψ(O) uun = 0.5 uun = 1 uun = 1.5 uun = 2 Figure 5.: Estimated performance metric ˆ Ψ(O)in scenario 1. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Uncertainty % -0.2 0 0.2 0.4 0.6 0.8 1 1.2 ˆ Ψ(O) uun = 0.5 uun = 1 uun = 1.5 uun = 2 Figure 6.: Estimated performance metric ˆ Ψ(O)in scenario 2. and the sensitiveness to faults. Further lines of research include: (i) decreasing the conservativeness of analysis/design using other types of Lyapunov functions, e.g. parameter-dependent ones; (ii) considering the case of noisy measurements and inexactly measured scheduling parameters; and (iii) integrating the proposed FDD approach with a fault tolerant control strategy. Funding This work has been supported by a grant from Iceland, Liechtenstein and Norway through the EEA Financial Mechanism. Operated by Universidad Complutense de Madrid (ref. 006-ABEL-IM-2014B). The authors also acknowledge the support by MINECO and FEDER through the project CICYT HARCRICS (ref. DPI2014-58104-R), by AGAUR through the contracts FI-DGR 2014 (ref. 2014FI B1 00172) and FIDGR 2015 (ref. 2015FI B2 00171), by the DGR of the Generalitat de Catalunya (ref. 2014/SGR/374) and by the Research Council of Norway through the Centres of Excellence funding scheme (ref. 223254 - AMOS). D. 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