Fault Detection for

Aircraft Control Surfaces Using

Approximate Input Reconstruction

Haoyun Fu, Jin Yan, ChenweiZhang, Dennis S. Bernstein

Department of Aerospace Engineering

The University of Michigan

Ann Arbor, MI 48109-2140

dsbaero@umich.edu

Harish J. Palanthandalam-Madapusi

Department of Mechanical and Aerospace

Engineering

Syracuse University

149 Link Hall

Syracuse, NY 13244

hjpalant@syr.edu

Abstract

We use an approximate input reconstruction algorithm to reconstruct unknown in-puts, which are then used as a basis for fault detection. The approximate inputreconstruction algorithm is a least squares algorithm that estimates both the un-known initial state and input history. The estimated inputs are then compared tothe commanded values and sensor values to assess the health of actuators and sen-sors. This approach is applied to the longitudinal and lateral dynamics of an aircraft.The input reconstruction algorithm can be used for minimum-phase or nonminimum-phase zeros; however, zeros on the unit circle yield persistent estimation errors andthus poor input reconstruction.

This work was sponsored by NASA under IRAC grant NNX08AB92A and the Aerospace Engineering

Department of the University of Michigan.

1 Introduction

As systems in general–and control systems in particular–become more complex,there is an increasing need to monitor critical components to ensure their properoperation. This problem is known generically as either fault diagnosis or fault detec-

tion [1–5].

A widely studied approach to input reconstruction is to use input observers toreconstruct inputs to the system based on an available system model and measure-ments. The estimated inputs can then be compared to expected inputs to assess thehealth of sensors and actuators. The relevant literature on this topic has its roots insystem inversion theory developed for either input observers (left inversion) or pre-view control (right inversion) [6–17]. One of the difficulties in system inversion is thepresence of zeros. If the system has no zeros, then input reconstruction is possibleeven if the initial state is zero [17]. However, if the system has zeros, then thereexists an initial state such that, for some nonzero input, the output is identicallyzero. Therefore, exact input reconstruction is impossible if the system has at leastone zero. For a minimum-phase zero, the unobservable input decays, and thus asymp-totically input observation is possible. On the other hand, for a nonminimum-phasezero, the unobservable input increases without bound, and thus input observation ispossible asymptotically backward in time, that is, noncausally for large data sets. Asa consequence, if a system has a minimum-phase zero and a nonminimum-phase zero,then input reconstruction is possible for time steps that are neither large nor small.Finally, if the zero lies on the unit disk, then the unobservable input is persistent(non-decaying in either forward or backward time) and thus input reconstruction isnot feasible.

In the present paper we begin by reviewing the exact input reconstructionmethod developed in [17]. Since this approach assumes that the system has no zeros,we estimate the input by using a least squares solution of the input-output relation.The resulting estimate is unable to estimate input components in the null space ofthe coefficient matrix, which corresponds to the unobservable input component.

We apply this algorithm to the linearized longitudinal and lateral dynamics of anaircraft in order to estimate thrust inputs as well as control-surface deflections. Theseestimates can be used to detect faults in the actuator or linkage sensor associated withthe control surface. We consider examples in which the dynamics are minimum phaseand nonminimum phase, as well as strictly proper and exactly proper, the latteroccurring when the measurement is given by an accelerometer.

1

2 Faults in an Aircraft Dynamic System

We consider the aircraft elevator and engine subsystems, which can potentiallyundergo various faults. The elevator is assumed to have a local sensor, called thelinkage sensor, which provides a measurement of the deflection of the linkage thatdrives the elevator. Consequently, the elevator subsystem can undergo various faults;for example, the control surface may be damaged, the linkage may be damaged, orthe linkage sensor may malfunction.

Figure 1 shows the various signals that are used for fault diagnosis. Specifically,δek,com is the command to the elevator, δek is the reconstructed (estimated) value ofthe elevator deflection, and lk,msmt is the signal from the linkage sensor. As discussedbelow, discrepancies between these signals suggest the possibility of various types offailures. Figure 2 shows the analogous signals for engine fault diagnosis.

Notation Descriptionδek,com commanded elevator inputδek,act actual elevator outputlk,act actual linkage outputlk,msmt linkage sensor measurementδek estimated elevator input

δTk,com commanded engine inputδTk,act actual engine outputδTk,msmt actual RPM sensor output

δTk estimated engine input

Table 1: Notation for Signals Used to Identify Faults

We consider the possibility that the linkage can fail in various ways. In particu-lar, we consider linkage faults that include saturation, rate saturation, and deadzone.These cases are defined in Table 2. In addition, linkage sensor faults include bias,drift, frozen, and calibration error. These faults are listed in Table 3, where b is aconstant and c is a positive number.

The elevator control surface can fail in various ways. For example, the elevatormay become stuck in a single, fixed deflection. Alternatively, the elevator may becomedeformed or damaged, resulting in a loss of control effectiveness. Detection of thesefaults is relevant to health monitoring to avoid loss of control system performance.

For the elevator, we detect faults during flight by comparing the commandedinput δek,com, the measured inputs from the linkage sensor lk,msmt, and the estimatedinputs from input reconstruction δek. Table 4 shows the logic by which these signalsare compared. For example, if the linkage measurement differs from the commandedelevator deflection and the estimation elevator deflection, then we can conclude that

2

Linkage Faults Description

Saturation lk,act =

{

δek,com, |δek,com| ≤ a

a, |δek.com| > a(a > 0)

Rate Saturation lk,act =

{

δek,com, |δek.com − δek−1,com| ≤ a

δek−1,com + a, |δek,com − δek−1,com| > a(a > 0)

Deadzone lk,act =

δek,com + a, δek,com ≤ −a0, −a < δek,com < a

δek,com − a, δek,com ≥ a

(a > 0)

Frozen lk,act = a (a ∈ R)

Table 2: Linkage Faults

Linkage Sensor Fault DescriptionBias lk,msmt = lk,act + b

Drift lk,msmt = lk,act + kc

Frozen lk,msmt = b

Scale Factor Error lk,msmt = lk,actb

Table 3: Linkage Sensor Faults

the linkage sensor is faulty but that the linkage and elevator are both operational.In practice, all of these signals are noisy, and thus error criteria are needed. Forsimplicity, Table 1 is stated in terms of equality and inequality of signals. A similarfault analysis can be applied to other control surfaces and the engine.

Case Condition Linkage Sensor Linkage Elevator

1 δek,com = δek = lk,msmt Operational Operational Operational2 δek,com = δe6 = lk,msmt Faulty Operational Operational3 δek,com 6= δek = lk,msmt Operational Faulty Operational4 δek,com = lk,msmt 6= δek Operational Operational Faulty

5δek,com 6= δek andδek,com 6= lk,msmt

Combination of Cases 2, 3, 4

Table 4: Fault Detection Analysis for the Elevator

3 Input Reconstruction

In this section, we briefly review the input reconstruction method developedin [17]. We consider both the strictly proper and exactly proper cases, which are usedin later sections for numerical examples.

3

3.1 Strictly Proper Case

Consider the linear discrete-time system

xk+1 = Axk +Hek, (3.1)

yk = Cxk, (3.2)

where xk ∈ Rn, ek ∈ R

p, yk ∈ Rl, A ∈ R

n×n, H ∈ Rn×p, and C ∈ R

l×n. The input ek

and the initial state x0 are assumed to be unknown. Without loss of generality, weassume l ≤ n, rank(C) = l > 0, and rank(H) = p > 0. For a nonnegative integer r,define Yr ∈ R

(r+1)l and Er ∈ R(r+1)p as

Yr△=

y0

y1...yr

, Er△=

e0e1...er

. (3.3)

Definition 3.1. Let r ≥ 1. Then the input and state unobservable subspace Ur

of (3.1), (3.2) is the subspace

Ur△=

{[

x0

Er−1

]

∈ Rn+rp : Yr = 0

}

. (3.4)

Definition 3.2. The system (3.1), (3.2) is input and state observable if Ur = {0}for all r ≥ r0.

Define Γr ∈ R(r+1)l×n, Mr ∈ R

(r+1)l×rp, and Ψr ∈ R(r+1)l×(n+rp) by

Γr△=

C

CA

CA2

...CAr

, Mr△=

0 0 . . . 0CH 0 . . . 0CAH CH . . . 0

......

. . ....

CAr−1H CAr−2H . . . CH

, (3.5)

and

Ψr△=

[

Γr Mr

]

. (3.6)

Then from (3.1), (3.2), we can write

Yr = Γrx0 +MrEr−1 = Ψr

[

x0

Er−1

]

, (3.7)

so that

Ur = N(Ψr), (3.8)

4

where N denotes null space. Next, define the positive integer

r0△=

{

max{⌈ n−ll−p

⌉, 1}, p < l,

1, p = l,(3.9)

where ⌈a⌉ denotes the smallest integer greater than or equal to a. Note that r0 is notdefined in the case p > l.

Theorem 3.3. The following statements are equivalent:

1. (3.1), (3.2) is input and state observable.

2. For all r ≥ r0, Yr = 0 if and only if

[

x0

Er−1

]

= 0.

3. For all r ≥ r0, rank(Ψr) = n+ rp.

4. There exists r ≥ r0 such that rank(Ψr) = n + rp.

5. rank(Ψn−1) = n + (n− 1)p.

Theorem 3.3 shows that (3.1), (3.2) is input and state observable if and only ifΨr has full column rank for all r ≥ r0. In this case the unique solution of (3.7) is

[

x0

Er−1

]

= Ψ†rYr, (3.10)

where † represents the Moore-Penrose generalized inverse

Ψ†r = (ΨT

r Ψr)−1ΨT

r . (3.11)

3.2 Exactly Proper Case

Consider the linear discrete-time system

xk+1 = Axk +Hek, (3.12)

yk = Cxk +Gek, (3.13)

where G ∈ Rl×p, while A, H, C, xk, ek, and yk are defined as in (3.1), (3.2). Without

loss of generality, we assume l ≤ n, rank(C) = l > 0, and rank

[

H

G

]

= p > 0. Due

to Gek, the output yk is directly affected by ek as well as by the past values of ek.Therefore, we have

Yr = Ψr

[

x0

Er

]

, (3.14)

5

where Er is defined by (3.3), Ψr△=

[

Γr Mr

]

∈ R(r+1)l×[n+(r+1)p], and

Mr =

G 0 · · · 0 0CH G · · · 0 0

......

. . ....

CAr−2H CAr−3H · · · G 0CAr−1H CAr−2H · · · CH G

. (3.15)

Furthermore, we have the following definition.

Definition 3.4. Let r ≥ 0. Then the input and state unobservable subspace Ur

of (3.12), (3.13) is the subspace

Ur△=

{[

x0

Er

]

∈ Rn+(r+1)p : Yr = 0

}

. (3.16)

The input and state unobservable subspace is given by Ur = N(Ψr). Next, ifp < l, then define

r0△= ⌈ n

l−p⌉ − 1. (3.17)

Since n > l − p, it follows that r0 ≥ 1.

Definition 3.5. The system (3.12), (3.13) is input and state observable if Ur ={0} for all r ≥ r0.

Theorem 3.6. The following statements are equivalent:

1. (3.12), (3.13) is input and state observable.

2. For all r ≥ r0, Yr = 0 if and only if

[

x0

Er

]

= 0.

3. rank(Ψr) = n + (r + 1)p for all r ≥ r0.

4. There exists r ≥ r0 such that rank(Ψr) = n + (r + 1)p.

5. rank(Ψn−1) = n(p + 1).

If (3.12), (3.13) is input and state observable, then Theorem 3.6 implies thatΨr has full column rank for all r ≥ r0. In this case the unique solution of (3.14) is

[

x0

Er

]

= Ψ†rYr, (3.18)

where

Ψ†r = (ΨT

r Ψr)−1ΨT

r . (3.19)

6

4 Invariant Zeros and Unobservable Inputs

If a linear system system has invariant zeros, then it is not input and stateobservable, that is Ur 6= {0}. In particular, there exists an initial state and a nonzeroinput such that the output is identically zero. Therefore, it is not possible to exactlyreconstruct the initial state and the input vector from the measured output. Never-theless, we use (3.10) and (3.18) for approximate input reconstruction, although thegeneralized inverses of Ψr and Ψr are no longer given by (3.11) and (3.19). We callthis the approximate input reconstruction method.

Approximate reconstruction must account for the locations of system transmis-sion zeros in the complex plane relative to the unit disk, as shown in Figure 3. Ifall of the transmission zeros are contained in the open unit disk, then approximatecausal reconstruction is possible for sufficiently large data sets and large times k. Onthe other hand, if all of the transmission zeros are contained in the complement of theclosed unit disk, then approximate noncausal reconstruction is possible for sufficientlylarge data sets and small times k. Consequently, if all of the system transmission zerosare contained in either the open unit disk or the complement of the closed unit disk,then approximate noncausal reconstruction is possible for sufficiently large data setsand small times k. Finally, if at least one transmission zero lies on the unit circle, thenapproximate reconstruction is not possible since a persistent reconstruction error willcorrupt the reconstructed inputs.

Let Pr denote the orthogonal projector onto Ur, and let Pr⊥△= I − Pr denote

the orthogonal projector onto U⊥r . Therefore, for the strictly proper case we have

Pr = Ψ†rΨr (4.1)

and similarly for the exactly proper case. We thus have the following definition.

Definition 4.1. The observable state and input are given by

[

x0,rec

Er−1,rec

]

△= Pr

([

x0

Er−1

])

, (4.2)

while the unobservable state and input are given by

[

x0,unrec

Er−1,unrec

]

△= Pr⊥

([

x0

Er−1

])

. (4.3)

It follows from Definition 4.1 that

x0 = x0,rec + x0,unrec, (4.4)

Er−1 = Er−1,rec + Er−1,unrec. (4.5)

7

Therefore, if the linear system is not input and state observable, the initial state

and input are consist of the components

[

x0,rec

Er−1,rec

]

and

[

x0,unrec

Er−1,unrec

]

. Therefore,

the accuracy of the initial state and input reconstruction depends on the magnitude

of the unobservable components

[

x0,unrec

Er−1,unrec

]

. As numerical examples show, the

reconstructed input Er−1,unrec is small for small and large values of k, respectively,in a system that has only minimum phase and nonminimum-phase zeros. In bothcases the reconstruction process involves a batch-processing algorithm. However, inthe nonminimum-phase case, the input estimates have good accuracy only for smallvalues of k, and thus the input reconstruction is noncausal.

4.1 Illustrative Example

To illustrate the unobservable state and input, we consider a minimal realizationof the transfer function

F (z) =z − a

(z − 0.1)(z − 0.2)(z − 0.3), (4.6)

where the zero a is given by a = 0.1, a = 10, a = 1, or a = −1. For the case a = 0.1,Figure 4 shows that the reconstruction improves asymptotically as k increases, butis poor for small values of k. This behavior is characteristic of a minimum-phase zerofor which the unobservable component of the input converges to zero. In contrast,for the case a = 10, Figure 5 shows that the reconstruction degrades asymptoticallyas k increases, but is good for small values of k. This behavior is characteristic of anonminimum-phase zero for which the unobservable component of the input diverges.Consequently, the input reconstruction is noncausal.

For the case a = 1, Figure 6 shows that the reconstruction is poor for all valuesof k since the unobservable component of the input is constant and nonzero. Asshown in Figure 7, the same situation occurs for a = −1, where the unobservablecomponent of the input is oscillating. In both cases, the unobservable component ispersistent, and thus the input reconstruction is poor.

5 Application to Longitudinal Flight Dynamics

In this section we use the approximate input reconstruction algorithm to es-timate the input signals to a longitudinal flight dynamics model. In particular, weestimate the elevator deflection and thrust using measurements of range, altitude,pitch-angle perturbation, and total-velocity perturbation, which are assumed to be

8

available from flight sensors. The matrix Ψ†r is based on the state-space matrices of

the flight dynamic model. We consider the linearized F-16 model developed in [18],which provides the state-space matrices A, B, C, and D for both decoupled longitudi-nal and lateral flight dynamics once the altitude and the total velocity of the aircraftare specified.

For longitudinal case, the state-space form of the F-16 model is

X = AX +BU, (5.1)

Y = CX +DU, (5.2)

where X△=

[

x h θ u α q]T

, U△=

[

δT δe]T

, and Y△=

[

x h u]T

. Thestates are range x (ft), altitude h (ft), pitch-angle perturbation θ (deg), total-velocityperturbation u (ft/s), angle of attack α (deg), and pitch rate q (deg/s). The inputsare elevator deflection δe from trim and thrust δT , while the available measurementsare x, h, and u. The matrices A ∈ R

6×6, B ∈ R6×2, C ∈ R

3×6, and D ∈ R6×2 are

given by

A =

0 0 0 1 0 00 0 5 3.553e− 10 −5.000e+ 2 00 0 0 0 0 10 1.080e− 04 −3.217e+ 01 −1.328e− 02 −7.326 −1.1960 2.076e− 06 2.547e− 13 −2.553e− 04 −6.398e− 01 9.378e− 010 2.573e− 20 0 −3.163e− 18 −1.568 −8.791e− 01

,

(5.3)

B =

0 00 00 0

1.565e− 03 7.397e− 02−2.445e− 07 −1.357e− 03

0 −1.137e− 01

, (5.4)

C =

1 0 0 0 0 00 1 0 0 0 00 0 0 1 0 0

, D = 0. (5.5)

For the system (5.1)–(5.5), we apply the approximate input reconstructionmethod. The condition number of Ψr is 2.8901e + 11 for r = 100, which impliesthat (5.1)–(5.5) is input and state observable. We obtain the state-space matrices forthe trim condition at an altitude of 15,000 ft with a velocity of 500 ft/s. The initial

9

state is thus set to be x0 =[

0 15000 0 500 0 0]T

. The thrust command δT

is chosen to be a constant, and the elevator deflection command δe is chosen to besinusoidal. White measurement noise with a signal to noise ratio of 10 is added toeach measurement in this and all subsequent examples. Since D = 0, the unknowncommands are estimated using (4.2) for r = 100.

Figure 8 shows the commanded inputs and their estimates, including both theobservable and unobservable components. The unobservable component is close tozero, which is consistent with the observation that the system is input and stateobservable.

Fault detection is conducted for a scenario in which the engine is given twosuccessive step commands, while the elevator becomes stuck at t = 45 sec. We usethe same initial state, commanded inputs, measurement noise, and step command, asabove. Figure 9 shows commanded inputs, measured inputs from the linkage sensorand estimated inputs both before and after the fault occurs.

Alternatively, we assume that x, h, and θ are the available outputs, and thus

C =

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 0

. (5.6)

The condition number of Ψr in this case is 2.6326e+14 for r = 100,which suggests that(5.1)–(5.5) is not input and state observable. With the same initial state, commands,and measurement noise, Figure 10 shows the the unknown commanded inputs andtheir estimated inputs. The unobservable component is large, which is consistentwith the computed condition number.

6 Application to Lateral Flight Dynamics

We now consider the lateral dynamics of the linearized F-16 model, which aregiven by

X = AX +BU, (6.1)

Y = CX +DU, (6.2)

whereX =[

φ ψ u β p r]T, U =

[

δT δa δr]T

, and Y =[

φ ψ u β]T

.The states are roll angle φ (deg), yaw-angle perturbation ψ (deg), total-velocity per-turbation u (ft/s), sideslip angle β (deg), roll rate p (deg/s), and yaw rate r (deg/s),

10

which the inputs are thrust δT (lb), aileron deflection δa (deg), and rudder deflec-tion δr (deg) from trim, while the available measurements are φ, ψ, u, and β. Thematrices A ∈ R

6×6, B ∈ R6×3, C ∈ R

4×6, and D ∈ R6×3 are given by

A =

0 0 0 0 1 7.810e− 020 0 0 0 0 1.0030 0 −1.328e− 02 9.403e− 15 0 0

6.414e− 02 0 3.761e− 20 −2.022e− 01 7.827e− 02 −9.919e− 010 0 4.496e− 18 −2.292e+ 01 −2.254 5.408e− 010 0 −7.040e− 18 6.005 −4.040e− 02 −3.146e− 01

,

(6.3)

B =

0 0 00 0 0

1.565e− 03 0 00 1.724e− 04 5.058e− 040 −4.623e− 01 5.686e− 020 −2.437e− 02 −4.687e− 02

, (6.4)

C =

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 1 0 0

, D = 0. (6.5)

For the system (6.1)–(6.5), we apply the approximate input reconstructionmethod with r = 100. The condition number of Ψr is 7.8790e + 7, which impliesthat (6.1)–(6.5) is input and state observable. We obtain A, B, C, and D for thetrim condition at an altitude of 15000 ft with a velocity of 500 ft/s. The initial state

is set to be x0 =[

0 0 500 0 0 0]T

. The thrust command δT is chosen to bea square wave, the aileron deflection command δa is chosen to be a sawtooth wave,and the rudder deflection command δr is chosen to be sinusoidal. All measurementsare assumed to be corrupted by white noise with SNR = 10. Since D = 0, the un-known commands are estimated using (4.2) for r = 100. During the flight, the enginedevelops a saturation fault at t = 53 sec, and the ailerons develop a deadzone att = 49 sec. Also the linkage sensor of the rudder is assumed to develop a scale-factorerror at t = 51 sec, which can be determined by comparing the actual command andestimated command with the measured command obtained from the linkage sensor.Figure 11 shows the fault detection results for the engine and control surfaces.

11

7 Application to Nonminimum Phase, Exactly Proper

Flight Dynamics

We now consider a nonminimum-phase flight example, in which the dynamicsare exactly proper. We require an inertial Earth frame FE with an arbitrary origin,whose axes ıE and E are horizontal, and whose axis kE points downward. The aircraftframe is fixed to the body of the aircraft, and its origin OAC is located at the aircraftcenter of mass c. The frame origin OAC, along with the frame vectors ıAC and kAC,are shown in Figure 12.

We assume that an accelerometer is installed at the location p on the aircraftin order to measure acceleration apz in the direction kAC. The position vector fromthe aircraft center of mass c to p is lıAC, and thus |l| is the distance from the aircraftcenter of mass c to p. For a typical business jet with velocity of 675.12 ft/sec atan altitude of 4000 ft [19], it can be verified from aircraft kinematics that the trimcondition is open-loop stable and that the transfer function from the output of theaccelerator to the elevator deflection input is

Gapz/δe(s) =acz(s)

δe(s)(7.1)

= −(42.14 − 17.65l)s4 + (11923.70− 26.11l)s3 + (88.46 − 0.12l)s2 + 79.20s

s4 + 2.01074s3 + 8.04799s2 + 0.084973s+ 0.067893.

This system is nonminimum phase as long as l < 276. In addition, the dynamicsare exactly proper case except when the coefficient of the highest order of s in thetransfer function equals zero, which occurs when l = 2.388.

To apply approximate input reconstruction to fault detection, we assume thatthe elevator command is a square wave and that the elevator becomes stuck. Theelevator deflection is estimated using (3.18) for r = 100. The measured output isthe acceleration in the direction kAC is corrupted by white noise with signal to noiseratio SNR = 10. Figures 13 and 14 show the unknown deflection inputs and theirestimates in the presence of noise with l chosen to be 50 and 274, respectively. Boththe observable and unobservable components of the input signals are shown. It canbe seen that, for l = 50, the unobservable component of the input is close to zero,whereas, for l = 274, the unobservable component increases with time, which isconsistent with the fact that the aircraft dynamics have a nonminimum-phase zero.

12

8 Conclusions

As an extension of exact input reconstruction, we considered an approximateinput reconstruction method where a least squares solution is used in the pres-ence of zeros. For minimum phase zeros, the input-reconstruction error decays; fornonminimum-phase zeros it grows; and for zeros on the unit circle, it is persistent.We applied this technique to aircraft fault detection, where the objective is to com-pare the estimated input with the commanded input and the linkage sensor. Thiscomparison provides a technique for online fault detection. We demonstrated thetechnique for both lateral and longitudinal dynamics, as well as nonminimum-phaseexactly proper dynamics. Future work will focus on the effect of model errors andsensor noise on the fault-detection accuracy.

9 Acknowledgments

We thank Siddharth Kirtikar for helpful comments.

References

[1] R. K. Douglas and J. I. Speyer. Robust fault detection filter design. Proc. Amer.

Contr. Conf., 91–96, 1995.

[2] R. Isermann, Fault Detection and Diagnosis in Engineering Systems. CRC, 1998.

[3] D. Dasgupta, K. KrishnaKumar, D. Wong, and M. Berry, “Negative SelectionAlgorithm for Aircraft Fault Detection,” in Proceedings of Third International

Conference on Artificial Immune Systems, pp. 1-13. Springer, 2004.

[4] R. Isermann, Fault Diagnosis Systems: An Introduction form Fault Detection to

Fault Tolerance. Springer, 2005.

[5] M. Blanke, M. Kinnaert, J. Lunze, and M. Saroswiecki, Diagnosis and Fault-

Tolerant Control. Springer, 2006.

[6] J. L. Massey and M.K. Sain. Inverses of linear sequential circuits. IEEE Trans-

actions of Automatic Control, C-17:330–337, 1968.

[7] M. K. Sain and J. L. Massey. Invertibility of linear time-invariant dynamicalsystems. IEEE transactions on automatic control, AC-14:141–149, 1969.

[8] M. Tomizuka. Zero phase error tracking algorithm for digital control. ASME

Journal of Dynamic Systems, Measurement, and Control, 109:65–68, 1987.

13

[9] E. Gross and M. Tomizuka. Experimental flexible beam tip tracking controlwith a truncated series approximation to uncancelable inverse dynamics. IEEE

transactions on control systems technology, 2:382–391, 1994.

[10] E. Gross, M. Tomizuka, and W. Messner. Cancellation of discrete time unstablezeros by feedforward control. Journal of Dynamic Systems, Measurement, and

Control, 116:33–38, 1994.

[11] J. Chen, R. J. Patton, and H.-Y. Zhang. Design of unknown input observers androbust fault detection filters. Int. J. Contr., 63:85–105, 1996.

[12] M. Hou and R. J. Patton. Input observability and input reconstruction. Auto-

matica, 34:789–794, 1998.

[13] G. Marro, D. Prattichizzo, and E. Zattoni, “A unified algorithmic setting forsignal-decoupling compensators and unknown-input observers,” Proc. Conf. Dec.

Contr., pp. 4512–4517, Sydney, Australia, 2000.

[14] G. Marro, D. Prattichizzo, and E. Zattoni, “Convolution proiles for right inver-sion of multivariable non-minimum phase discrete-time systems,” Automatica,Vol. 38, pp. 1695–1703, 2002.

[15] S. Gillijns. Kalman Filtering Techniques for System Inversion and Data Assim-

ilation. PhD thesis, 2007.

[16] J. A. Butterworth, L. Y. Pao, and D. Y. Abramovitch. The effect ofnonminimum-phase zero locations on the performance of feedforward model-inverse control techniques in discrete-time systems. 2008 American Control Con-

ference, 2008.

[17] H. J. Palanthandalam-Madapusi and D. S. Bernstein, A Subspace IdentificationAlgorithm for Simultaneous Identification and Input Reconstruction. Proc. Conf.

Dec. Contr., pp. 4956–4961, New Orleans, LA, December, 2007; Int. J. Adapt.

Control Signal Process, to appear.

[18] R. S. Russell, Non-linear F-16 Simulation using Simulink and Matlab, June,2003. <http://www.aem.umn.edu/people/faculty/balas/darpa_sec/software>.

[19] J. Roskam, Airplane Flight Dynamics and Automatic Flight Controls. DARcor-poration, 2001.

14

Figure 1: Signals used for elevator fault detection

15

Figure 2: Signals used for engine fault detection

16

Figure 3: Regions of invariant zeros and their effect on approximate input reconstruc-tion

17

0 5 10 15 20 25 30 35 40 45 500.99994

0.99996

0.99998

1.00000

1.00002

Time

Inpu

t

InputEstimated Input

Figure 4: Example (4.6) with minimum phase zero a = 0.1. The unobservable com-ponent is close to zero except for small values of k.

0 5 10 15 20 25 30 35 40 45 50−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Time

Inpu

t

InputEstimated Input

Figure 5: Example (4.6) with nonminimum-phase zero a = 10. The unobservablecomponent is close to zero except for large values of k.

18

0 5 10 15 20 25 30 35 40 45 500

0.2

0.4

0.6

0.8

1

1.2

Time

Inpu

t

InputEstimated Input

Figure 6: Example (4.6) with zero a = 1 on the unit circle. In this case, the recon-structed input has a persistent component.

0 5 10 15 20 25 30 35 40 45 500.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

Time

Inpu

t

InputEstimated Input

Figure 7: Example (4.6) with zero a = −1 on the unit circle. In this case, thereconstructed input has a persistent component.

19

0 10 20 30 40 50 60 70 80 90 1001999.5

2000

2000.5

Time (sec)

Thr

ust (

lb)

Commanded InputEstimated Input

0 10 20 30 40 50 60 70 80 90 100−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Time (sec)

Ele

vato

r D

efle

ctio

n A

ngle

(de

g)

Figure 8: Input reconstruction for F-16 longitudinal flight. Estimates of the unknowninputs δT and δe are obtained by using measurements of the outputs x, h, and u andthe linearized flight model. In this case, the unobservable input is close to zero.

0 10 20 30 40 50 60 70 80 90 1001700

2100

2500

2900

3300

Time (sec)

Thr

ust (

lb)

Commanded InputMeasured InputEstimated Input

0 10 20 30 40 50 60 70 80 90 100−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Time (sec)

Ele

vato

r D

efle

ctio

n A

ngle

(de

g)

Figure 9: Input reconstruction for F-16 longitudinal flight. Estimates of the unknowninputs δT and δe are obtained by using measurements of the outputs x, h, and u andthe linearized longitudinal flight model. For this example, the input reconstructionalgorithm indicates that the engine is operational, whereas the elevator is stuck.

20

0 10 20 30 40 50 60 70 80 90 1001999.6

1999.8

2000

2000.2

2000.4

2000.6

2000.8

Time (sec)

Thr

ust (

lb)

Commanded InputEstimated Input

0 10 20 30 40 50 60 70 80 90 100−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Time (sec)

Ele

vato

r D

efle

ctio

n A

ngle

(de

g)

Figure 10: Input reconstruction for F-16 longitudinal flight. Estimates of the un-known inputs δT and δe are obtained by using measurements of the outputs x, h,and u and the linearized flight model. In this case, the unobservable input is not closeto zero, suggesting the presence of an invariant zero near the unit circle.

0 10 20 30 40 50 60 70 80 90 100

1999

2000

2001

Time (sec)

Thr

ust (

lb)

Commanded InputMeasured InputEstimated Input

0 10 20 30 40 50 60 70 80 90 100−0.1

−0.05

0

0.05

0.1

Time (sec)

Aile

ron

Def

lect

ion

Ang

le (

deg)

0 10 20 30 40 50 60 70 80 90 100−0.1

−0.05

0

0.05

0.1

Time (sec)

Rud

der

Def

lect

ion

Ang

le (

deg)

Figure 11: Input reconstruction for F-16 lateral flight. Estimates of the unknowninputs δT, δa, and δr are obtained by using measurements of the outputs φ, ψ, u,and β, as well as the linearized lateral flight model. For this example, the inputreconstruction algorithm indicates that the engine thrust is saturated, the aileronshave a deadzone nonlinearity, and the rudder has a faulty sensor.

21

Figure 12: Earth and aircraft frames. The forward-mounted accelerometer locationis located at the point p, whose distance from the center of mass c is given by |l|.

22

0 10 20 30 40 50 60 70 80 90 100−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Time (sec)

Ele

vato

r D

efle

ctio

n A

ngle

(de

g)

Commanded InputMeasured InputEstimated Input

Figure 13: Nonminimum-phase aircraft example with forward-mounted accelerome-ter. For this example, the distance from the aircraft center of mass to the accelerom-eter is l = 50 ft. The output is acz, while δe is the input. Both observable andunobservable components are shown. Note that the unobservable component is closeto zero, which indicates the absence of zeros near the unit circle.

0 10 20 30 40 50 60 70 80 90 100−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Time (sec)

Ele

vato

r D

efle

ctio

n A

ngle

(de

g)

Commanded InputMeasured InputEstimated Input

Figure 14: Nonminimum-phase aircraft example with forward-mounted accelerome-ter. For this example, the distance from the aircraft center of mass to the accelerom-eter is l = 275 ft. The output is acz, while δe is the input. Both observable andunobservable components are shown. Note that the unobservable component is largefor large k, which indicates the presence of a nonminimum-phase zero close to theunit circle.

23