Sliding mode observers for robust fault estimation in linear parameter varying systems

A linear parameter varying sliding mode fault reconstruction scheme is proposed. Compared with other conventional sliding mode observer designs, the estimate emerges quite naturally from the equations governing the sliding motion, and although the process may be viewed as a filtering of the discontinuous injection, there is no phase lag introduced in the process. A semi‐analytic solution to the proposed linear parameter varying sliding mode observer is first discussed. This motivates the structure of the linear matrix inequality formulation which is proposed, and clarifies the class of systems to which the approach is applicable. The linear matrix inequality problem set up minimizes the induced ℒ2$$ {\mathcal{L}}_2 $$ gain between the fault estimate and external disturbances, whilst ensuring linear quadratic performance of the underpinning sliding motion. The proposed scheme has been validated using the Japan Aerospace Exploration Agency's Multi‐Purpose Aviation Laboratory aircraft‐in‐the‐loop platform to demonstrate the design efficacy.

information to cancel the effect of the disturbance on the system outputs of interest.Another area where unknown input reconstruction is also employed is in the creation of virtual sensors.In many engineering systems certain quantities of interest cannot be directly measured-either because of physical in-accessibility, or else the absence of a sensor capable of measuring the quantity of interest directly.In these situations virtual sensors can be developed whereby other quantities are measured which, together with a model of the system capturing the interaction of the unmeasurable quantities and those which are measured, can be posed as an unknown input estimation problem. 5,13,14Finally the fast developing area of protecting systems against cyber-attacks can also be posed in an unknown input problem framework. 15,16ecause of the engineering applicability of the unknown input reconstruction problem for linear time invariant (LTI) systems, unsurprisingly, a rich collection of different approaches have been developed.For example the ubiquitous (Extended) Kalman filter 17,18 has been used-often involving augmenting the state-space representation with the unknown input signals considered to be states (under the assumption the rate of the change of the unknown input is small).Corless and Tu 19 developed a well cited bespoke formulation based on a linear observer formulation for LTI systems.Other more recent work by Zak and co-workers 20 also investigated this problem.In an FDI setting Stoustrup and Neimann 21 are considered to be the first to address the problem of robust fault estimation using an  ∞ methodology.The  ∞ method for fault estimation has subsequently been considered extensively in aerospace problems. 22,23igh-gain observers 24 have been presented in a number of contexts, which typically involves augmenting the states with the unknown input signal and creating an augmented system in which the unknown input is the derivative of the disturbance signals*.Many of the approaches originally developed in an LTI setting have been extended for LPV plant representations. 2 LPV systems have been widely exploited in an aerospace problem setting as a formal extension to gain scheduling.Some of the fault estimation work based on  ∞ methods has been extended to LPV system representations. 9lso in recent years, LTI unknown input observers have been extended into a linear parameter varying framework 25,26 and the state affine framework. 27liding mode observers were first investigated and heralded in terms of their robustness that is, estimating the unknown internal states from measured inputs and outputs in the presence of disturbances.However in the 1990's it was demonstrated that by appropriate processing of the equivalent output error injection signal necessary to maintain sliding, the unknown input (faults) could be simultaneously estimated along with the states. 22,28,29Subsequently this sparked interest in the use of sliding mode observers for fault detection and estimation. 30The early work of Walcott and Zak 20 and Edwards and Spurgeon 31 consider conventional sliding mode observers and were subject to relative degree one and minimum phase conditions on the dynamical system relating the unknown input to the measurement.Work attempting to obviate these conditions appears in Reference 30.(Minimum phase and relative degree one conditions are also related to the classical constraints for linear UIOs.)The relative degree condition was completely obviated by the use of higher-order sliding mode observers/differentiators as proposed by Levant; 32 Fridman 33 combined these with a Luenberger observer framework to create a general form of unknown input observer.The ideas have also been extended to nonlinear systems. 34Higher order sliding mode observers have been also extended to create finite time sliding mode observers. 35,36These ideas have also been applied to various aerospace applications such as the 3-DoF helicopter, 37 a quadrotor UAV 35 and multi-sensor fusion problems. 38Higher order sliding modes effectively address the problem of unknown input reconstruction in non-unity relative degree systems.However in sensor fault reconstruction problems, relative degree is not the primary difficulty (such problems are relative degree zero in nature).As a consequence conventional sliding mode observers can be employed and model robustness issues are addressed at the point of design.In these formulations, the unknown input signal is reconstructed via the equivalent output error injection signal. 39Traditionally this involves the use of low pass filters.This helps remove the effects of noise but adds a phase delay to the reconstruction.The methodology presented in this article allows the fault estimate to emerge naturally from the dynamics associated with the sliding motion.As a consequence, although filtering is still involved, there is no additional phase delay introduced in the unknown input estimation signal.
In this article, the approach is applied to a sensor fault reconstruction problem.Specifically the proposed novel sliding mode observer scheme is validated using the Japan Aerospace Exploration Agency's (JAXA) Multi-Purpose Aviation Laboratory (MuPAL-) aircraft-in-the-loop (AIL) platform, incorporating a full scaled manned aircraft and a pilot.The validation results show promising fault reconstruction performance, with and without wind/gust, in the face of various fault scenarios.
The remainder of the article is structured as follows: in Section 2, the formulation of the observer involving the synthesis of the observer gains (via an LMI optimization) and the modulation gain is discussed; the AIL platform, testing scenarios, and AIL validation results are presented in Section 3. Finally, Section 4 provides some concluding remarks.

OBSERVER FORMULATION
Consider an nth order uncertain LPV system affected by a class of sensor faults modeled as where the measured output y ∈ R p and the unknown sensor fault f ∈ R q with n > p > q.The system input u ∈ R m and the external (unknown) disturbance d ∈ R k .This lumped term is intended to capture the modeling mismatch between the nominal LPV representation in (1) and the true engineering system being monitored.Here N ∈ R p×q is the fault distribution matrix which is assumed to be rank q.In ( Remark 1.In analytic fault redundancy approaches the fidelity of model about which the observer is designed is important.Improved modeling accuracy helps minimize false alarms and improves fault reconstruction accuracy.The move from LTI to LPV models is a natural way to achieve this-especially in aerospace applications. Remark 2. In this article it is assumed (t) is perfectly known.This is a common assumption in the LPV literature.However, some researchers have investigated how imperfect knowledge of (t) affects the performance of observers. 40These also include a class of LPV SMOs. 41,42nsider the application of a nonsingular (output) scaling matrix where N 2 ∈ R q×q is nonsingular.From the rank assumption on N, the matrix Q in (2) can be obtained by Gaussian elimination (or QR decomposition).
Since the output distribution matrix C = QC associated with the scaled outputs is also full row rank there always exists a nonsingular square matrix T c ∈ R n×n and the change of coordinates x  → T c x that creates a system representation in which the output distribution matrix has the form . Hence, w.l.o.g it is assumed the representation in (1) is such that Define y as the scaled outputs y according to y = Qy and partition y as col(y 1 , y 2 ) then where . The objective is to develop an LPV observer for the system in (1) based on the scaled outputs y 1 and y 2 of the form where the injection signal  is a function of y 2 − C 2 z.The aim is to induce sliding on y 2 − C 2 z for the purpose of estimating the fault signal f .The objective is to minimize the effect of the plant model mismatch by minimizing the effect of d in (1) on the fault reconstruction error in an induced  2 norm sense.The next section develops a possible analytic solution to give an insight into the problem, and to provide the necessary conditions under which the problem is solvable.

An analytic solution
Here it is assumed that (A(), C) is quadratically observable (Assumption 1).Define a further change of coordinates (x 1 , x 2 )  → (x 1 , x 2 ) according to x 1 = x 1 + Lx 2 where L ∈ R (n−p)×p .As argued in Reference 43 there exist choices of L such that the system can be written as where Ā11 () is quadratically stable and x 2 = y.Define x 2 = col(x 21 , x 22 ), where x 21 ∈ R p−q and x 22 ∈ R q , then the scaled outputs are Note by definition x 21 is a known quantity (because it is obtained from y) and it is free from corruption that is, independent of f .Consequently the system in ( 5)-( 6) can be written in partitioned form as for appropriate choices of submatrices.For notational convenience write ( 9)- (11) as and where Define a (sliding mode) observer for the system (12)-( 14) based on (4), and taking account of the change of coordinates, as where z = col(z 1 , z 21 , z 22 ) is partitioned conformally with the partition of x in ( 9)- (11).The signal  in (15) represents a nonlinear injection signal depending on (y 2 − z 22 ) whose purpose is to produce a sliding motion in finite time in the estimation error space.Define the observer gain matrix in (15) associated with  as Also define G 1 () and G 2 () as and where A s 22 ∈ R (p−q)×(p−q) and A s 33 ∈ R q×q are Hurwitz matrices which constitute design freedom.This will be shown to represent a semi-analytic solution to the problem.In particular, assume that A s 33 is symmetric negative definite.The injection signal in (15) is defined as where the so-called modulation gain (t) will be defined explicitly later.
In particular and then substituting from ( 9)-( 11) and (15), it follows that Assuming  can be selected to enforce sliding, during sliding, e y 2 = y 2 − z 22 ≡ 0. It follows from ( 20) that ē22 = −N 2 f .From (21), the sliding motion is determined by where ē11 = col(ē 1 , ē21 ).During sliding e y 2 = ̇ey 2 = 0 so consequently from the last row of ( 21) since ̇ē 22 = −N 2 ̇f where  eq is the equivalent output error injection 39 necessary to maintain sliding.Defining f = −N 2 f it follows that the equations above can be written as and that Define where in the notation of Reference 2 that is, from filtering the discontinuous injection  eq through the operator W().In the situation when d = 0 in (24) and W() (which is necessary for fault estimation), then f = f holds for all  ∈ Ω.The stability of the W() is ensured during the design process which will be discussed in the sequel.
Remark 3. Note that f is a continuous signal as it emerges from W() despite the fact that  eq is discontinuous.Critically (26) may be viewed as an explicit way of precisely designing a low pass filter to extract the equivalent injection (in this case f ).Here 1st order SMOs are being employed but a continuous estimate f is obtained.Further, there is no artificially (or additionally) implemented filter to formally extract the equivalent output error injection which is used to create the estimate f .More generally, in the practical situation where d ≠ 0, from ( 24) and ( 25), the signals d and the fault estimation error f − f are connected via the operator H(): specifically where Note that in ( 27) and ( 29), the dynamics are both governed by where Ā(), F(), C 1 , and C 2 are defined in ( 12)-( 14).If Ā0 () is (quadratically) stable for all  ∈ Ω and d ≡ 0, then perfect reconstruction of f is achieved asymptotically by f , despite nonzero initial conditions and changes in operating conditions (i.e., changes in (t)).
Remark 4. Theoretically once the effects of any initial state estimation error have decayed, f → f and so there is no phase delay between f and f .This is in contrast to previously proposed schemes where f is extracted from the discontinuous injection signal via low pass filtering which results in bandwidths above which phase distortion becomes inevitable.Of course in a practical engineering situation, other unavoidable elements such as discretization, the properties of the hardware A/D and D/A employed to implement the FDI schemes, and latency in the calculations and so forth.will contribute a delay (which will not be mitigated in this proposed approach).
If Ā0 () is quadratically stable for all  ∈ Ω, W() and H() can be considered as stable low pass filters.For the purpose of inducing robustness of the fault estimation error f − f against the effect of the external disturbance d, the design freedom in the observer given by L needs to be chosen to minimize the induced  2 norm of H() in (29).Notice that the operators H() and W() can be realized in the original coordinates of ( 1)-(3) as and F(), M() are defined in (1).
A linear matrix inequality (LMI) based approach will be presented in the next section to synthesize the observer gains numerically.

Synthesis of the observer gains
The previous section developed insight into the necessary conditions for developing the observer in (4).Write the system in (1) in the form where A 11 () ∈ R (n−q)×(n−q) and A 22 () ∈ R q×q .(Note this x = col(x 1 , x 2 ) partition is of different dimension to the one in Section 2.1 and independent of the developments in that section.)The scaled outputs This section will reverse-engineer the choice of observer gains G 1 () and G n , assuming G 1 () in (4) has been chosen so that is quadratically stable.This development starts in the original coordinate system based on the scaled outputs y.For this approach to be applicable, the pair (A() − F()N −1 2 C 2 , C 1 ) must be detectable.An insight into this is given in the Appendix A.
Assuming the design freedom G 1 () has been chosen so that A 0 () in ( 34) is quadratically stable and that the corresponding Lyapunov function † is in particular and let x = Tx.The matrix P 11 is guaranteed to be invertible since P is s.p.d.In the new x coordinates where and therefore has block diagonal form.Define where ] and C2 = and where where in particular Ȃ11 () = A 11 () + P − where B() = TB() and define Gn = Then in the error coordinates ȇ = x − z if ȇ = col(ȇ 1 ȇ2 ) the error system where M() = TM().Note that the sliding motion is governed by the system matrix Ȃ11 () which is guaranteed to be quadratically stable from (47). Define It follows from (47) that the system matrix Ȃ11 () governing sliding satisfies To allow tuning of the performance during the sliding mode, consider the matrix inequality For given designer selected weights W 1 and W 2 , part of the synthesis can be formulated as the LMI problem min This has the form of an LQG observer design problem for the fictitious pair (A 11 (), A 21 ()) for a specific choice of cost function involving the weights W 1 and W 2 .The formulation can also be viewed as the natural dual of the commonly exploited quadratic cost approach for sliding mode control hyperplane design. 39,44he other design objective is to minimize the induced  2 norm of H() from (29).Minimizing the  2 gain of H() can also be cast as an LMI problem min where Y () = PG 1 () as defined earlier, and  denotes an upper bound on the induced  2 norm of H().
The overall design problem can be cast as min P,Y () 1 tr(P −1 11 ) +  2  subject to (55) and (57), where  1 and  2 are scalar weightings.This process synthesizes the gain G 1 () directly.From the s.p.d matrix P, the components P 11 and P 12 can be identified.This specifies the matrix T in (38) and therefore from (51) LPV systems admit various representations, such as the polytopic/affine LPV representation, 45 the gridding-based LPV representation synthesis 2,46 and the multiplier-based linear fractional transformation (LFT) representation. 47,48To solve (55), (57), and (58) tractably involving a finite number of LMIs "separating" the parameter-dependent system matrices and the Lyapunov matrix, here the following concession is introduced: Assumption 2. It is assumed that A(), F(), and M() depend affinely on .In particular where the parameter  i is the ith component of .
Using the vertex property discussed in Reference 45, the LMIs in ( 55) and ( 57) can be converted into a polytopic form and the synthesis of the observer gain G 1 () and the analysis of the design problem in (58) is undertaken only at the vertices of the polytope.
Remark 5.The above analysis is based on the establishment of a (fixed) quadratic Lyapunov function in (35).This introduces conservatism but avoids a situation in which an infinite number of matrix inequalities need to be solved.
The choice of the modulation gain to guarantee sliding is discussed in the next subsection.

Design of the modulation gain
To develop a theory for selecting the modulation gains to enforce sliding in (51), consider the signal together with the dynamics from ( 51) T , then the operator G  (s) mapping w  →  given by ( 61) and ( 60) is quadratically stable.Therefore since by assumption ||d(t)|| < k d and ||f (t)|| < k f , for any initial condition ȇ1 (0) at time t = 0, in finite time t 0 , there exist positive scalars  0 , m 1 , m 2 and m 3 such that for all t ≥ t 0 . 24The scalars  0 , m 1 , m 2 , and m 3 will now be used in the definition of the modulation gain.
Proposition 1.By choosing the modulation gain in (19) according to where  is a positive design scalar, a sliding motion on y 2 − z 22 = 0 is guaranteed in finite time.
Proof.Define the output estimation error e y 2 ∶= y 2 − z 22 .Using the fact that e y 2 = ȇ2 + N 2 f and it is easy to verify from ( 51) and a sliding motion is guaranteed in finite time.▪ Remark 6.The modulation gain in (63) depends on knowledge on the bounds of d, f , and ̇f .Consequently the choice of (t) requires a certain level of engineering judgment, knowledge and understanding of the system for which the monitoring scheme is being designed.To remove the assumption that bounds on d, f and ̇f need to be known, adaptive schemes can be used.A recent possible choice is the "dual-layer" approach. 49e observer discussed in this section will now be applied to the problem of reconstruction of a yaw rate sensor fault associated with the MuPAL- aircraft.

MuPAL-𝜶 platform
The MuPAL- Aircraft-in-the-loop (AIL) platform 50 is currently used by the Japan Aerospace Exploration Agency to validate the performance of user-defined advanced guidance and control laws and FDD/FTC methodologies for both ground AIL tests inside the hanger (see Figure 1), 51,52 and actual flight tests including Simple Adaptive Control, 53  ∞ Control, 23,[54][55][56][57] Sliding Mode Control (SMC) 11,44 and an Indirect Adaptive Control strategy involving online parameter estimation. 58he schematic of the configuration of the MuPAL- AIL test platform is shown in Figure 1A.Clearly, this configuration is different from the traditional flight simulator and it includes onboard actuators as components of the loop.Clearly from Figure 1A, a Fly-By-Wire (FBW) system is involved in the AIL schematic, which allows advanced flight control laws and validation scenarios to be created flexibly by the users.Figure 1A also includes an emulation computer which contains a high-fidelity aircraft motion model (typically of the Dornier aircraft motion model itself).Hence the model outputs (the aircraft's position and attitude), can be obtained from the emulation computer, and the outputs and the actual control surface deflections are used by the FBW computer to update the controller command, the cockpit displays and the outside view on the graphic display screen located outside the cockpit (see Figure 1D).All the control surfaces respond to the signals sent from the FBW system, and the actual position of the surfaces are used as inputs into the aircraft motion model in the emulation computer.
According to JAXA's protocols, all user-defined flight control laws are written in C-code which contains both the logic of the flight control law and the interface between the flight control and the FBW system.After the compilation, the codes are "burned" into the FBW computer.The "flight data" obtained from the AIL is recorded for the purpose of performance analysis.

Observer design results
This article specifically focuses on the aircraft's lateral-directional dynamics.The scheduling parameters are chosen as where v eas represents the equivalent airspeed. 56Because the regulatory constraints in JAXA mean most test flights are conducted at similar altitudes with almost identical mass and center of gravity positions.The scheduling parameters are v eas and its square.Here the conservative assumption that  1 (= v eas ) and  2 (= v 2 eas ) are independent of each other was made and they are normalized to lie in the interval [0 1].The LPV system and input matrices are provided in Appendix B.
The system states of the LPV model are given by which denote roll angle, sideslip angle, roll rate and yaw rate, respectively.Here it is assumed that the system outputs are and therefore The system inputs for the purpose of these tests are where  a and  r represent deflections of the aileron and the rudder.Consequently, in this example, n = 4, p = 3, and q = 1.Since only the yaw rate measurement is assumed to be corrupted by faults, the fault distribution matrix F() = 0 and The disturbance distribution matrix M() is assumed to be fixed and to take the form which captures the effect of external disturbances such as wind and gusts impacting on the roll rate and yaw rate channels.
The design freedom in ( 55) are chosen as W 1 = 0.1 and W 2 = 0.5I 3 .From Assumption 2, the LMIs in ( 55) and ( 57) are affine with respect to  and they can be synthesized using the SEDUMI solver with YALMIP parser.The weightings are chosen as  1 = 0.2 and  2 = 0.8, which gives a higher priority to the disturbance rejection.The observer gains G 1 () and G 2 () are in the form of and their individual components are −11.0614 5.0103 0.0000 0.0000 −13.0727 −0.0000 63.6898 −3.3304 −0.0000 −0.0000 122.3122 0.0000 In this design, the values of L are which ensures that Ā11 () is quadratically stable.The tuning matrix Ȃ * 22 = −0.1 and the modulation gain  is chosen as 0.2.For safety reasons, it was determined that the magnitude of the sensor faults would be less that 5 deg/s.This is the maximum level of sensor faults typically tested during a flight evaluation campaign. 56The magnitude of wind/gusts is assumed to be less than 4 m/s (This is also typical of the sort of conditions which may be encountered during flight testing.).Since (t) is bounded in a known compact set, the values of m 1 , m 2 , and m 3 in (62) can be estimated from ( 60) and (61).Then the initial modulation gain is chosen to ensure a sliding motion.The final value of (⋅) was decided upon following extensive pre-flight ground testing on the MuPAL- aircraft configured in an AIL setup.
The next section describes the implementation and testing of the observer scheme on JAXA's AIL.

AIL test results
During the AIL tests, the flaps and gear are set to up in the emulation model, and the Direct Lift Control (DLC) FLAPS are fixed at 0 deg.The AIL test results in the following subsections are associated with a selected flight condition in which the mass of the aircraft is 5700 kg, the center of gravity is at 28%, the trim altitude is maintained at 5000 ft and the trimmed indicated airspeed is around 120 kts.The baseline controller, taken from Reference 44, is used to ensure good sideslip and roll angle tracking performance.During the AIL tests, the sideslip and roll angle commands are Autopilot commands.
During the AIL tests, the yaw rate sensor measurement is corrupted by faults created at a software level.The faults which will be explored are typical of those used to test the efficacy of FDI schemes in the aerospace community. 4In Reference 59, the particular SMO was benchmarked against several metrics including false alarms, miss detections and fault detection times, and so forth

Fault free without wind/gust
First the results from a fault free AIL test without wind/gusts are shown in Figure 2.This scenario is used to provide a performance benchmark.An "S-turn" manoeuvre was created by the Auto-pilot and the response of the lateral directional states are shown in Figure 2A. Figure 2A demonstrates that sideslip and roll angle tracking performance is good.The corresponding aileron and rudder deflections are plotted in Figure 2B.Next, the nominal observer performance is shown in Figure 2C.In this baseline test, Figure 2C (bottom left) demonstrates the fault reconstruction error.For the majority of the test the error was in the bound ±0.25 deg/s.Clearly there is correlation between the loss of accuracy in terms of fault estimation and the times at which the manoeuvres are initialized and finished.The worst case deviations in accuracy are all between ±0.5 deg/s (Figure 9A). Figure 2C also shows sliding is maintained throughout.Though the comparisons between the histogram in the fault free case (see Figure 9A) with those in faulty cases, it will be seen that the best possible level of performance is obtained in this scenario since there are no wind and gust disturbances injected-although there are still un-modeled effects impacting on the results because of the hardware in the loop (the control surface actuator dynamics considered in Reference 56 have been ignored, there is latency in the communication etc).
Remark 7. To avoid false alarms, a simple thresh-holding approach would require a bound of at least ±0.5 to be utilized.This would be the main factor in determining the detection times.Of course more sophisticated thresh-holding mechanisms can be employed-see for example the adaptive scheme used in conjunction with the SMO in Reference 59.

Faulty cases-Sine wave
Figures 3 and 4 show the AIL evaluation results when an additive sine wave fault occurs on the yaw rate sensor from 30 s onwards.During this set of AIL tests, wind/gusts have been injected, but the same manoeuvre as the one created in the fault free scenario has been considered.The results obtained from two different sine wave faults are shown in this subsection.One sine wave fault signal has a frequency of 0.05 Hz and the other one has a frequency of 0.5 Hz.The amplitude of the two sine waves is 5 deg.In principle the low frequency one should be reconstructed more accurately and should not display any visible phase lag in the reconstruction.Figures 3 and 4 (top left) show that the additive sine wave faults appear in the yaw rate sensor measurements from 30 s onwards.Both faults can be accurately reconstructed (Figures 3 and 4 top right) and in both situations sliding is maintained (Figures 3 and 4 bottom left and right).Again in Figure 3 (and Figure 4) there is correlation between loss of accuracy, and the start/end of the manoeuvres.For most of the test time, the reconstruction errors lie between ±0.5 deg/s although there are a couple of deviations outside this bound.The standard deviation of the error plots for this test is larger than the one associated with the fault free case (Figure 9C).In Figure 4, the reconstruction error lies mostly within the ±0.5 deg/s bound as in the case with the lower frequency sine wave fault.The error signal in this case contains higher frequency components compared with the lower frequency sine wave, and the standard deviation is slightly larger (Figure 9C).This is perhaps to be expected because this is a more challenging situation compared to the lower frequency sine wave faulty case.

Faulty cases-Drift
Next, results (in the presence of wind/gusts) are shown when drift faults occur on the yaw rate sensor from 30 s onwards.Both a slow drift (with a rate of 0.017 deg ∕s 2 ) and a fast drift rate (0.025 deg ∕s 2 ) are considered as shown in Figures 5  and 6.During this set of AIL tests, no manoeuvre was created, but the testing period was increased to 5mins.The results correspond to the maximum allowable amplitude of fault which is permitted in actual flight tests. 11learly, although yaw rate sensor drifts are present, the fault estimation scheme can estimate the fault with a good level of precision.As shown in Figure 9D, the error distributions of both the drift scenarios are very similar to each other (and similar to the fault free case).The errors predominantly fall within the ±0.5 deg/s bound once more.Furthermore, Figures 5 and 6 (bottom right) show that sliding is maintained.

3.3.4
Faulty cases-NRZ Figure 7 is associated with a non-return-to-zero sensor fault which is commonly used by the aerospace industry to validate fault diagnosis performance in the face of discontinuous faults. 60The fault is set to occur from 30 s onwards.The switching rate of the NRZ signal is chosen to be 0.5 Hz, and the amplitude set to be 2 deg.Here the same doublet manoeuvre created by the Auto-pilot is considered for consistency.Figure 7 (top right) shows the fault estimation performance.This is the most challenging case in some ways.Although the frequency of 0.5 Hz is the same as the fast sinusoid scenario-now the fault is non-differentiable at the transition points.Thus formally, this class of faults does not meet the assumption of  Section 2. However this fault is used within the aerospace community to test the efficacy of FDI schemes 4,59 .Crucially, the points of non-differentiably are isolated and countable, and manifest themselves as sharp peaks in the fault estimation errors occurring when, instantaneously, sliding is broken.Whilst most of the time the errors remain between ±0.5 rad/s there are more pronounced tails to both sides of the error distributions as shown in Figure 9B.

3.3.5
Faulty cases-"Lock-in-the-place" Figure 8 shows the AIL evaluation results, in the presence of wind/gusts, when a "lock-in-the-place" occurs on the yaw rate sensor.For consistency, the manoeuvre created by the Auto-pilot is same as those used in previous tests.Figure 8A,B show the results when a "lock-in-the-place" occurs at 8 and 20 s, respectively.It can be seen from Figure 8A,B that, after the fault occurs, the yaw rate sensor measurements are frozen at fixed values.From Figure 8A,B (top right and bottom left), it can be seen that the faults are reconstructed accurately in the presence of wind/gusts.The error distribution here (Figure 9D) is very similar to the fault free case with the errors lying between ±0.5deg/s.Furthermore, sliding can be maintained when the "lock-in-the-place" occurs (Figure 8A,B bottom right).

RMS indices
Table 1 shows the performance indices of the proposed fault estimation scheme in the face of various fault scenarios.Here the root mean squared (RMS) values of the fault estimation errors are used to measure the precision of fault estimation.The relative size of the RMS values in Table 1 are consistent with the shape of histograms shown in Figure 9.The results largely accord with intuition.The poorest performance is associated with the NRZ fault-this is to be expected since formally this class of fault does not meet the differentiability requirements, and sliding is temporarily broken at times when the fault derivative does not exist.Best performance is achieved during a slow drift.Such an incipient fault is often difficult to detect until the level of fault exceeds a threshold above the typical levels of noise and disturbances experienced by the (fault free) systems.Sine wave faults are more challenging than drift faults to replicate and tend to expose lags in reconstruction.Also higher frequency sine waves are more challenging than lower frequency ones.These are borne out in the RMS values.

CONCLUSIONS
This article has considered the problem of unknown input estimation in a system which has a linear parameter varying structure.The article has proposed a scheme, based on classical sliding mode ideas which creates a smooth estimate of the unknown input.Although the process may be viewed as a filtering of the discontinuous injection, there is no additional phase lag introduced in the process.A clear design strategy has been articulated and the problem has been posed in terms of linear matrix inequalities: the objective is to minimize the  2 gain between the fault estimate and a class of external disturbance impacting on the system, whilst ensuring linear-quadratic regulator-like dynamics associated with the sliding motion.The proposed scheme was validated using the Japan Aerospace Exploration Agency's Multi-Purpose Aviation Laboratory (MuPAL-) aircraft-in-the-loop platform.Some aircraft-in-the-loop validation results were presented to demonstrate the design efficacy.
Remark 8.The condition that the (A  − F  N −1 2 C 2 , C 1 ) needs to be detectable is identical to the conditions which would be required if the SMO approach adopted in Reference 11 was employed to address the fault estimation problem in (1).

APPENDIX B
The constant matrices used to construct A() and B() are given by

F I G U R E 2 3 5 6
Fault-free case, no wind/gust.(A) The trajectories of the system states, (B) control surface deflections, (Faulty case, sine wave (frequency-0.05),wind/gust Time (s) Time (s) Time (s) Time (s)F I G U R E 4 Faulty case, fast sine waves (frequency-0.5),wind/gust Faulty case, slow drift, wind/gust, long run, no maneuver Faulty case, fast drift, wind/gust, long run, no maneuver

F I G U R E 7 F I G U R E 8 F I G U R E 9
Faulty case, NRZ, frequency 0.5, NRZ magnitude 2 deg/s, no wind/gust Faulty case, "lock-in-the-place", fault occurrence time (A) 8 s and (B) 20 s, wind/gust.Normalized histogram.(A) Fault free case, (B) NRZ, (C) sine wave, (D) drift, (E) lock-in-the-place

2 C 2 𝜂 1 2 ]= 0 ⇔
Future work could focus on the relaxation of the design conservatism via: (a) considering a parameter dependent Lyapunov matrix; (b) formulating the parameter dependent matrices A(), F(), and M() in a more general way; and (c) addressing the robustness issue with respect to certain classes of parametric uncertainty.and since N 2 is square and invertible, (A7) is equivalent to 2 = N −1 (sI − A  ) 1 + F  N −1 2 C 2  1 = 0 and − C 1  1 = 0. (A8)Thus R(s) has full column rank for all s ∈ C + if and only if 1) u and y are measurable, and it is assumed the output distribution matrix C has full row rank.Also assume that both f , d and ̇f are bounded and that specifically ||f || < k f , ||d|| < k d and || ̇f || < l f where the positive scalars k f , k d and l f are known.In this article, it is assumed the time-varying, differentiable scheduling parameter (t) ∈ Ω ⊂ R r is known accurately and belongs to a compact set Ω.
Assumption 1.It is assumed that (A(), C) is quadratically observable.
1 11 P 12 A 21 () − G 11 ()C 11 − P −1 11 P 12 G 12 ()C 11 is quadratically stable.Now consider the observer (in the x coordinates) as RMS values of the fault estimation errors TA B L E 1