Fault detection for non-linear system with unknown input ... - IEEE Xplore

www.ietdl.org Published in IET Signal Processing Received on 5th June 2012 Revised on 21st November 2012 Accepted on 4th February 2013 doi: 10.1049/iet-spr.2012.0171

ISSN 1751-9675

Fault detection for non-linear system with unknown input and state constraints Zhen Luo, Huajing Fang Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China E-mail: [email protected]

Abstract: This study extends the problem of fault detection (FD) for linear discrete-time systems with unknown input to non-linear systems. Moreover, based on physical consideration, the constraints of state are considered. A non-linear recursive filter is developed where the constrained state and the input are interconnected. Constraints which can improve the quality of estimation are imposed on individual updated sigma points as well as the updated state. The advantage of algorithm is that it is able to incorporate arbitrary constraints on the states during the estimation procedure. Unknown input which can be any signal is obtained by least-squares unbiased estimation and the state estimation problem is transformed into a standard unscented Kalman filter problem. By testing the mean of the innovation process, a real-time FD approach is proposed. Simulations are provided to demonstrate the effectiveness of the theoretical results.

1

Introduction

As the modern industry and science and technology develop rapidly, system becomes large, complex and intelligent. Once fault occurs, loss will be inestimable. In order to avoid big damage, a science of fault detection (FD) and diagnosis is established. Since FD technique is essential to improve the safety and reliability of dynamic systems, recently more and more attention has been paid to FD. Many FD methods have been developed to detect and identify sensor and actuator faults, such as analytical redundancy [1–3], a neural network [4], parameter identification method based on Fourier transform [5], testing the covariance matrix of the innovation sequence [6] and so on. A common FD approach is to keep tracking residuals of measurement and compare them against a set threshold value. However, the threshold value is difficult to be confirmed. A man-made threshold value can be defined according to the character of system fault, or according to relevant theory. In terms of simple to develop, easy to understand, expert system is wide-spread utilised by FD. Obviously, there are some drawbacks with expert system: relies entirely on knowledge of subject matter experts, significant number of rules required, significant management overhead to keep knowledge base up to date, precise inputs required, no confidence limits supplied. In this paper, an approach based on Kalman filter innovation is used to detect faults. A real-time detect of faults affecting the mean of the innovation process is examined. This approach does not require a priori statistical characteristics of the faults, and the computation burden is not very heavy. Among various methods, unscented Kalman filter (UKF) is the most widely used approach to analyse the stochastic 800 & The Institution of Engineering and Technology 2013

non-linear system and show good performance in many cases. UKF as an improvement to extended Kalman filter (EKF) was proposed by Xiong et al. [7]. This method is based on the unscented transform technique, a mechanism for propagating mean and covariance through a non-linear transformation [8, 9]. The state vector is represented by a minimal set of carefully chosen sample points, called sigma points, which approximate the posterior mean and covariance of the Gaussian random variable with a second-order accuracy In contrast, the linearisation technique used in the EKF can only achieve first-order accuracy [10, 11]. Furthermore, compared with the previous work based on Kalman filtering, it has the following advantages [12, 13] † It is not necessary to compute the Jacobian matrices in UKF. A more accurate non-linear measurement model can be employed. † UKF is widely used in practice, ranging from muti-sensor fusion, target tracking, position determination, to training of neural networks. † Owing to carefully chosen sample points, we can have multiple, measurement hypotheses and obtain the best observation in each time step, according to the measurement probability distribution function, which is in contrast with single hypothesis in Kalman filtering. † It is more robust than tracking algorithms based on Kalman filter. However, in order to estimate stochastic non-linear system from the UKF, it is generally assumed that all system parameters, noise covariance and inputs are known. Unlucky, these assumptions do not hold in many situations, IET Signal Process., 2013, Vol. 7, Iss. 9, pp. 800–806 doi: 10.1049/iet-spr.2012.0171

www.ietdl.org the UKF may fail in the presence of parametric uncertainties, noise covariance uncertainties and unknown inputs. Recently, Kalman filter with unknown input has become a focus of increased attention and the use of state constraints has increased in practical engineering problems, both in linear and non-linear systems. A traditional way to deal with the unknown input is augmenting the state vector and unknown input, which requires some assumptions [14]. Kitanidis [15] proposes an optimal recursive state filter which is based on the assumption that no prior information about the unknown input is available, whose stability and convergence conditions were established by Darouach and Zasadzinsi [16]. Hsieh [17] used a two-stage Kalman filtering technique to estimate both the state and unknown input, and the authors of [18, 19] proved that the estimation of the unknown input in [20] is indeed optimal. Constraints on state values (which may be based on physical consideration) are often neglected because they do not fit easily into the structure of Kalman filter. A number of approaches have been developed to use these state constraints within the Kalman filter framework, for example, estimation-projection [21], system-projection [22] and a two-step projection [9]. Gain-projection [23, 24] methods can be used to find the optimal way to incorporate equality constraints. Another way to incorporate equality constraints is the measurement-augmentation Kalman filter, in which a perfect ‘measurements’ of the constrained quantity is appended to the physical measurements [25]. Also various algorithms have been developed for inequality constraints, such as moving horizon estimator [26, 27], unscented filtering algorithms [28] and probabilistic methods [29]. On the one hand, the literatures mentioned above which deal with unknown input and state estimation problems are only limited to linear discrete-time systems. Moreover, they do not consider the state constraints. On the other hand, constraint handing algorithms discussed above are only suitable for a certain constraint question. Owing to various limitations as discussed above, in this paper, a non-linear recursive filter is developed where the constrained estimation of the state and the input are interconnected. Constraints are imposed on individual updated sigma points as well as the updated state. The advantage of algorithm is that it is able to incorporate arbitrary constraints on the states during the estimation procedure. Compared with unconstrained estimation of state, constrained state estimation significantly improves the estimation accuracy of the filter. Least-squares unbiased estimation algorithm can be used to obtain unknown input. The state estimation problem is transformed into a standard UKF problem which can be easily solved. By testing the mean of the innovation process, a real-time FD approach is proposed. The structure of this paper is as follow. In Section 2, the problem is formulated and the structure of the non-linear recursive filter is presented. Section 3 gives least-squares unbiased solution of the unknown input and then the state estimation problem is solved by KF algorithm. FD is presented in Section 4. Section 5 presents some simulation results. Conclusions are given in Section 6.

2

Problem formulation

In this section, the principle of using classical UKF for estimation is summarised. Consider the general discrete IET Signal Process., 2013, Vol. 7, Iss. 9, pp. 800–806 doi: 10.1049/iet-spr.2012.0171

non-linear system xt+1 = f (xt , ut ) + Gt dt + wt

(1)

yt = Ct xt + vt

(2)

xL ≤ x ≤ xU q(x) ≤ 0

(3)

such that

g(x) = 0 where xt is the n-dimensional state vector of the system, the non-linear mapping f(†) is the assumed to be continuously differentiable with respect to xt, dt is the m-dimensional unknown input vector, yt is the p-dimensional vector the of measurement output, Ct is the measurement matrix of the system of order p × n, wt is the random n-dimensional system disturbance vector and vt is the random p-dimensional vector of measurement noise. xL and xU are the lower and upper bounds on the state vector, q(x) and g (x) represent inequality and equality constraints. The subscript t is the time index. Moreover, wt and vt are the uncorrelated zero mean Gaussian random vectors and their covariance matrices are Q ≥ 0 and R > 0, respectively. We assume that the following conditions are satisfied _

E[x0 ] = x 0 E[xt vTm ] = 0 E[xt wTm ] = 0 _

(4)

_

E[(x0 − x 0 )(x0 − x 0 )T ] = P0 _

where E[†] is the expectation operator, x is the expected value of x. Throughout the paper, we assume that rank Ct Gt−1 = rank Gt−1 = m, for all t [16]. When no prior knowledge about the dynamical evolution of unknown input is assumed to be available and no prior assumption is made, that is, the unknown input can be any signal. A constrained non-linear recursive filter which estimates both the system_ states xt and the input dt based on the initial estimate x 0 and the sequence of the measurements {y0, y1,…,yt} is developed. The procedure for implementing the constrained estimation based on UKF can be summarized as follows: 2.1

Prediction step

The UT is used to determine mean and covariance of the propagated states. A set of 2n + 1 sigma points are calculated as _

_

xt−1 ( + ) = [x t−1 , x t−1 +














(n + l)Pt−1 ]

(5)

the transformed set sigma points are obtained by propagating each sigma point through process model as

xi,t|t−1 ( − ) = f (xi,t−1 ( + ), ut )

(6)

The predicted mean and state prediction error covariance matrix without considering unknown input are then 801

& The Institution of Engineering and Technology 2013

www.ietdl.org _

the variance of x t|t .

computed as _

x t|t−1 ( − ) =

2n 

_

vci xi,t|t−1 ( − )

i=0

Pt|t−1 ( − ) =

2n 

vci (xi,t|t−1 (

_∗ x t|t

_

− ) − x t|t−1 ( − ))

i=0 _

_ x t|t

(8)

(xi,t|t−1 ( − ) − x t|t−1 ( − )) + Q

(14)

_

_

= x t|t−1 + Gt−1 d t−1 _∗

(15)

_∗

= x t|t + Kt (yt − Ct x t|t )

(16)

T

where Pt_− 1 is the error covariance matrix. Note that the 












_ notation x t−1 + (n + l)Pt−1 means that the vector x t−1 is add//subtracted to each column in the matrix (n + λ)Pt − 1. 2.2

_

d t−1 = Mt (yt − Ct x t|t−1 )

(7)

Update/estimation step

Constrained state can be obtained by update individual sigma point as well as the updated state. The following algorithm is able to incorporate arbitrary constraints on the states during the estimation procedure min (xi,t|t−1 ( + ) − xi,t|t−1 ( − ))

T

xi,t|t−1

(Pt|t−1 ( − ))−1 (xi,t|t−1 ( + ) − xi,t|t−1 ( − ))

(9)

where Mt and Kt still have to be determined. vm 0 = l/(n + l), vc0 = l/(n + l) + (1 − a2 + b), vmi = vci = l/2(n + l), i = 1,…,2n, λ = n(α2 − 1). ωi is a set of scalar weights, and n is the state dimension; the parameter α determines the _ spread of the sigma points around x and is usually set to 1e − 4 ≤ α ≤ 1. The constant β is used to incorporate part of the prior knowledge of the distribution of x, and for Gaussian distributions, β = 2 is optimal.

3

Input estimation and state estimation

In this section, we consider the estimation of the unknown input and calculate the optimal gain Kt. _

Defining the innovation y˜ t by y˜ t = yt − Ct x t|t−1 , then y˜ t = Ct Gt−1 dt−1 + Dt

subject to the following constraints

_

where Dt = Ct (f (xt−1 ) − x t|t−1 + wt−1 ) + vt

xL ≤ xi,t|t−1 ( + ) ≤ xU q(xi,t|t−1 ( + )) ≤ 0

(10)

g(xi,t|t−1 ( + )) = 0 xL ≤  q  g

2n 

d t−1 = Mt Ct Gt−1 dt−1 + Mt Dt

(11)

Mt Ct Gt−1 = Im

vci (xi,t|t−1 ( + ) = 0

the constrained state and constrained state estimation error covariance matrix without considering unknown input are then calculated as =

2n 

vci xi,t|t−1 ( + )

(12)

i=0

Pt|t−1 =

2n 

_

vci (xi,t|t−1 ( + ) − x t|t−1 )

(13)

i=0 _

(21)

Theorem 1: Let rankC_t Gt−1 = rankGt−1 = m hold and R˜ t be a positive define. Let x t|t be an unbiased estimation of xt, then Mt can be given by −1

−1

Mt = (FtT R˜ t Ft )−1 FtT R˜ t

definite. There Proof: Under the assumption that R˜ t is positive T exists an invertible matrix S˜ t satisfying S˜ t S˜ t = R˜ t . Equation (17) can be transformed to −1 −1 −1 S˜ t y˜ t = S˜ t Ct Gt−1 dt−1 + S˜ t Dt

T

_

Let x t|t be an unbiased estimation of xt, then x t|t−1 is an unbiased estimation of f (xt − 1) but biased estimation of xt because of unknown input. Therefore an unbiased estimation of dt can be obtained in_∗(14) and used to calculate an unbiased state estimation x t|t−1 in (15). In the final step, update states, Kt can be derived by minimising 802 & The Institution of Engineering and Technology 2013

(22)

where Ft = CtGt − 1 [19].

(xi,t|t−1 ( + ) − x t|t−1 ) + Q _

(20)

Noting that d t−1 is unbiased if and only if Mt satisfies



i=0

_ x t|t−1

(19)

_

+) ≤0

i=0 2n 

_



vci (xi,t|t−1 (

(18)

Substituting (17) in (14), yield

vci (xi,t|t−1 ( + ) ≤ xU

i=0 2n 

then E[Dt ] = 0, E[Dt DTt ] = Ct Pt|t−1 CtT + Rt = R˜ t

(17)

(23)

−1 So S˜ t Ct Gt−1 has full column rank, the least-squares (LS) _

solution d t−1 of (23) equals _

−1

−1

d t−1 = (FtT R˜ t Ft )−1 FtT R˜ t y˜ t

(24)

IET Signal Process., 2013, Vol. 7, Iss. 9, pp. 800–806 doi: 10.1049/iet-spr.2012.0171

www.ietdl.org Combing the defining of y˜ t with (14), (25) can be derived _

d t = Mt y˜ t

(25)

Compared (24) with (25), then (22) holds. Next, we calculate the optimal gain Kt, and the _ minimum-variance unbiased estimation x t|t of xt can be derived. _ Defining d˜ t = dt − d t , it follows from (20) and (21) that d˜ t = −Mt Dt Defining x˜ ∗t = xt −

_∗ x t|t ,

(26)

it follows from (1) (15) and (26) that _

x˜ ∗t = (In − Gt−1 Mt Ct )(f (xt−1 ) − x t|t−1 + wt−1 ) − Gt−1 Mt vt ∗ Pt|t

=

Process FD based on UKF

A common FD approach is to keep tracking residuals of measurement and compare them against a threshold value. However, the threshold value is difficult to be confirmed. A man-made threshold value can be defined according to the character of system fault, or according to relevant theory. In terms of simple to develop, easy to understand, expert system is wide-spread utilised by FD. Obviously there are some drawbacks with expert system: relies entirely on knowledge of subject matter experts, significant number of rules required, significant management overhead to keep knowledge base up to date, precise inputs required, no confidence limits supplied. By testing the mean of the innovation process, a real-time FD approach is proposed in this section. Let the measurement error at time step t be defined by

(27)

E[˜x∗t x˜ ∗T t ]

= (In − Gt−1 Mt Ct )Pt|t−1 (In − Gt−1 Mt Ct )T

_

et = yt − y t|t _

(28)

T + Gt−1 Mt Rt MtT Gt−1

It follows from (2) and (16) that

(34)

_∗

where y t|t = Ct x t|t . To detect sensor/actuator faults, it is convenient to use normalised innovation sequences ∗ T Ct + R)−1/2 et e˜ t = (Ct Pt|t

_

x˜ t = xt − x t|t = (In − Kt Ct )˜x∗t − Kt vt

(29)

It follows from (29) that the error covariance matrix Pt is given by ∗ Pt = E[˜xt x˜ Tt ] = (In − Kt Ct )Pt|t (In − Kt Ct )T + Kt Rt KtT (30)

Theorem 2: The gain matrix ∗ T ∗ T K t = Pt|t Ct (Ct Pt|t Ct + Rt )−1 [30]

4

Kt

is

given

(31)

Using the matrix differential formulas ddK tr[KA] = AT and T d dK tr[KAK ] = 2KA. Solving for Kt the result is ∗ T ∗ T Ct (Ct Pt|t Ct + Rt )−1 Kt = Pt|t

IET Signal Process., 2013, Vol. 7, Iss. 9, pp. 800–806 doi: 10.1049/iet-spr.2012.0171

bt =

t 

e˜ Tj e˜ j

(36)

j=t−M +1

where M is the number of the implementation (window length). This statistical function (36) has χ2 distribution with M degrees of freedom and is denoted by χ2M

bt  x2M If the hypothesis γ1 is correct then χ2 level for a confidence probability α will be greater than χ2 level found for the hypothesis γ0, that is

(32)

Substituting (32) in (30), the error covariance matrix can now be simplified as defined by (33) ∗ Pt = (In − Kt Ct )Pt|t

Since then E[˜et ] = 0 E[˜et e˜ Tj ] = Pe˜ = I d(tj), where δ(tj) is the Kronecker symbol, I is the identity matrix. Two hypotheses are introduced γ0: no sensor fault occurs; and γ1: a sensor fault occurs. To detect sensor/actuator faults changing the mean of the innovation sequence, the following statistical function may be used, see [31, 32]

by

Proof: The Kalman gain Kt, which minimises the mean-square error, can be found by different means but the most expedient is to differentiate with respect to Kt and set the differentials to zero to solve for Kt. This is done according to (31) d ∗ tr[Pt ] = −2(In − Kt Ct )Pt|t + 2Kt Rt = 0 dK

(35)

(33)

g0 :bt ≤ x2a,Ms

∀t

g1 :bt . x2a,Ms

∃t

803

& The Institution of Engineering and Technology 2013

www.ietdl.org 5

Simulation

In this section, a non-linear model of a single-link flexible joint robot system is described by the following set of [33]

u˙ m = vm v˙ m =

k B k k (ul − um ) − R vm + t u − t m + d Jm Jm Jm Jm

u˙ l = vl v˙ l =

k mgh (u − u m ) − sin (ul ) + d J1 l J1

where θm and ωm are, respectively, the position and angular velocity of DC motor and θl and ωl represent those of the link. u is the excitation signal. d represents an unknown input. We measure θm and θl. μ denotes a fault signal. System parameters are given as follows: The moment of inertia of the DC motor Jm = 3.7 × 10 − 3 kg m2, link intertia J1 = 9.3 × 10 − 3 kg m2, pointer mass m = 2.1 × 10 − 1 kg, link length h = 3 × 101 m, torsional spring constant k = 1.8 × 10 − 1 Nm/rad, viscous friction coefficient BR = 4.6 × 10 − 2 Nm/V, amplifier gain Kt = 8 × 10 − 2 Nm/V. Simulation the non-linear system with the unknown input d = sin(t) u = 1 M = 20 and α = 0.95. The initial state and _ associated covariance matrix are taken as x 0 = [0 0 0 0]T and P0 = 36 ∗ I4×4 , respectively. Also by constraining xi,t|t−1 ( + ) ≥ 0 the sigma points are restricted to physical valid values. In order to improve the accuracy of FD, the failure threshold value is defined as 5% less than the normal value x2a,Ms = 101.88. The fault can be judged if the mean of the innovation sequence exceed the 95% of the normal value. Case 1: No faults happen. The states as estimated by unconstrained filter and constrained filter approach are presented in Fig. 1. It can be seen from the figure that both approaches track the true state. However, the preciseness of constrained filter is higher than that of unconstrained filter. Fig. 2 shows that the actual input can be tracked well by the estimated unknown inputs. The statistic function is lower than the threshold which is given in Fig. 3. Moreover, the statistic function of constrained filter is lower than that of unconstrained filter. It means that the

Fig. 1 Actual and estimated states without fault 804 & The Institution of Engineering and Technology 2013

Fig. 2 Actual and estimated input without fault

Fig. 3 Without faults

Fig. 4 Actual and estimated states with fault IET Signal Process., 2013, Vol. 7, Iss. 9, pp. 800–806 doi: 10.1049/iet-spr.2012.0171

www.ietdl.org testing the mean of the innovation process, a real-time FD approach is proposed. This algorithm does not require a priori statistical characteristics of the faults, and is easy to be implement.

7

Fig. 5 Actual and estimated input with fault

Fig. 6 Sensor FD in the mean value

constrained estimation significantly improves the prediction accuracy of the filter. Case 2: The fault occurs at the step 40 and μ = 1.62. The simulation results in this case are given in Figs. 4–6. The second estimated state and input curves have deviated from actual state and input curves after iteration 40 in Figs. 4 and 5 because of the fault. The fault causes changes in the mean of the innovation sequence, after 10 steps and 15 steps, the statistic function of constrained filter and unconstrained filter exceed the threshold which is given in Fig. 6, respectively. That is, the fault can be detected by constrained filter is five steps earlier than unconstrained filter.

6

Conclusion

A non-linear filter based on UKF is developed which simultaneously estimates the input and the constrained state. Constraints which can improve the quality of estimation are imposed on individual updated sigma points as well as the updated state. Unknown input which can be any signal is obtained by LS unbiased estimation and the state estimation problem is transformed into a standard UKF problem. By IET Signal Process., 2013, Vol. 7, Iss. 9, pp. 800–806 doi: 10.1049/iet-spr.2012.0171

References

1 Zhang, Y.M., Li, X.R., Ieee: ‘Detection and diagnosis of sensor and actuator failures using interacting multiple-model estimator’, Proc. 36th IEEE Conf. on Decision and Control, (1997) 2 Maybeck, P.S.: ‘Multiple model adaptive algorithms for detecting and compensating sensor and actuator/surface failures in aircraft flight control systems’, Int. J. Robust Nonlinear Control, 1999, 9, (14), pp. 1051–1070 3 Rago, C., Prasanth, R., Mehra, R.K., Fortenbaugh, R., Ieee: ‘Failure detection and identification and fault tolerant control using the Imm-Kf with applications to the Eagle-Eye Uav’, Proc. 37th IEEE Conf. on Decision and Control, (1998) 4 Alessandri, A.: ‘Fault diagnosis for nonlinear systems using a bank of neural estimators’, Comput. Ind., 2003, 52, (3), pp. 271–289 5 Perhinschi, M.G., Lando, M., Massotti, L. et al.: ‘On-line parameter estimation issues for the Nasa Ifcsf-15 fault tolerant systems’, Proc. 20th Annual American Control Conf. (ACC), (2002) 6 Hajiyev, C., Caliskan, F.: ‘Sensor and control surface/actuator failure detection and isolation applied to F-16 flight dynamic’, Aircraft Eng. Aerospace Technol., 2005, 77, (2), pp. 152–160 7 Xiong, K., Zhang, H.Y., Chan, C.W.: ‘Performance evaluation of Ukf-based nonlinear filtering’, Automatica, 2006, 42, (2), pp. 261–270 8 Julier, S.J., Uhlmann, J.K.: ‘A new extension of the Kalman filter to nonlinear systems’, (1997) 9 Julier, S.J., LaViola, J.J.: ‘On Kalman filtering with nonlinear equality constraints’, IEEE Trans. Signal Process., 2007, 55, (6), pp. 2774–2784 10 Julier, S.J., Uhlmann, J.K.: ‘Unscented filtering and nonlinear estimation’, Proc. IEEE, 2004, 92, (3), pp. 401–422 11 Kandepu, R., Foss, B., Imsland, L.: ‘Applying the unscented Kalman filter for nonlinear state estimation’, J. Proces. Control, 2008, 18, (7–8), pp. 753–768 12 Li, P.H., Zhang, T.W., Ma, B.: ‘Unscented Kalman filter for visual curve tracking’, Image Vis. Comput., 2004, 22, (2), pp. 157–164 13 Dini, D.H., Mandic, D.P., Julier, S.J.: ‘A widely linear complex unscented Kalman filter’, IEEE Signal Process. Lett., 2011, 18, (11), pp. 623–626 14 Hsieh, C.S., Ieee: ‘Optimal minimum-variance filtering for systems with unknown inputs’, WCICA 2006: Sixth World Congress on Intelligent Control and Automation, Vol 1–12, Conf. Proc., 2006, pp. 1870–1874 15 Kitanidis, P.K.: ‘Unbiased minimum-variance linear state estimation’, Automatic, 1987, 23, (6), pp. 775–778 16 Darouach, M., Zasadzinski, M.: ‘Unbiased minimum variance estimation for systems with unknown exogenous inputs’, Automatica, 1997, 33, (4), pp. 717–719 17 Hsieh, C.S.: ‘Robust two-stage Kalman filters for systems with unknown inputs’, IEEE Trans. Autom. Control, 2000, 45, (12), pp. 2374–2378 18 Gillijns, S., De Moor, B.: ‘Unbiased minimum-variance input and state estimation for linear discrete-time systems with direct feedthrough’, Automatica, 2007, 43, (5), pp. 934–937 19 Gillijns, S., De Moor, B.: ‘Unbiased minimum-variance input and state estimation for linear discrete-time systems’, Automatica, 2007, 43, (1), pp. 111–116 20 Kerwin, W.S., Prince, J.L.: ‘On the optimality of recursive unbiased state estimation with unknown inputs’, Automatica, 2000, 36, (9), pp. 1381–1383 21 Simon, D., Chia, T.L.: ‘Kalman filtering with state equality constraints’, IEEE Trans. Aerosp. Electron. Syst., 2002, 38, (1), pp. 128–136 22 Ko, S., Bitmead, R.R.: ‘State estimation for linear systems with state equality constraints’, Automatica, 2007, 43, (8), pp. 1363–1368 23 Teixeira, B.O.S., Chandrasekar, J., Torres, L.A.B., Aguirre, L.A., Bernstein, D.S.: ‘State estimation for linear and non-linear equality-constrained systems’, Int. J. Control, 2009, 82, (5), pp. 918–936 24 Teixeira, B.O.S., Chandrasekar, J., Palanthandalam-Madapusi, H.J., Torres, L.A.B., Aguirre, L.A., Bernstein, D.S.: ‘Gain-Constrained Kalman Filtering for Linear and Nonlinear Systems’, IEEE Trans. Signal Process., 2008, 56, (9), pp. 4113–4123 25 Walker, D.M.: ‘Parameter estimation using Kalman filters with constraints’, Int. J. Bifurcation Chaos, 2006, 16, (4), pp. 1067–1078 26 Rao, C.V., Rawlings, J.B., Mayne, D.Q.: ‘Constrained state estimation for nonlinear discrete-time systems: stability and moving horizon approximations’, IEEE Trans. Autom. Control, 2003, 48, (2), pp. 246–258 805

& The Institution of Engineering and Technology 2013

www.ietdl.org 27 Rao, C.V., Rawlings, J.B., Lee, J.H.: ‘Constrained linear state estimation – a moving horizon approach’, Automatica, 2001, 37, (10), pp. 1619–1628 28 Vachhani, P., Narasimhan, S., Rengaswamy, R.: ‘Robust and reliable estimation via unscented recursive nonlinear dynamic data reconciliation’, J. Process Control, 2006, 16, (10), pp. 1075–1086 29 Rotea, M., Lana, C.: ‘State estimation with probability constraints’, Int. J. Control, 2008, 81, (6), pp. 920–930 30 Kalman, R.E.: ‘A new approach to linear filtering and prediction problem’, IEEE Trans. Autom. Control, 1960, 82, (Series D), pp. 35–45

806 & The Institution of Engineering and Technology 2013

31 Soken, H.E., Hajiyev, C.: ‘Pico satellite attitude estimation via robust unscented Kalman filter in the presence of measurement faults’, ISA Trans., 2010, 49, (3), pp. 249–256 32 Hajiyev, C.M., Caliskan, F., Inst Electr, E.: ‘Fault detection in flight control systems via innovation sequence of Kalman filter’, UKACC Int. Conf. on Control’98, Vols I&Ii, 1998, pp. 1528–1533 33 Sharma, R., Aldeen, M.: ‘Fault and disturbance reconstruction in non-linear systems using a network of interconnected sliding mode observers’, IET Control Theory Appl., 2011, 5, (6), pp. 751–763

IET Signal Process., 2013, Vol. 7, Iss. 9, pp. 800–806 doi: 10.1049/iet-spr.2012.0171