Automatica 45 (2009) 1896–1902

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Automatica

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Brief paper

Set-membership filtering for systems with sensor saturationI

Fuwen Yang a,b,∗, Yongmin Li ba School of Information Science and Engineering, East China University of Science and Technology, Shanghai 200237, PR Chinab Department of Information Systems and Computing, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom

a r t i c l e i n f o

Article history:Received 6 August 2008Received in revised form21 January 2009Accepted 5 April 2009Available online 27 May 2009

Keywords:Set-membership filteringSensor saturationUnknown-but-bounded noiseConvex optimisationState estimation ellipsoid

a b s t r a c t

This paper addresses the set-membership filtering problem for a class of discrete time-varying systemswith sensor saturation in the presence of unknown-but-bounded process and measurement noises. Asufficient condition for the existence of set-membership filter is derived. A convex optimisationmethod isproposed to determine a state estimation ellipsoid that is a set of states compatiblewith sensor saturationand unknown-but-bounded process and measurement noises. A recursive algorithm is developed forcomputing the ellipsoid that guarantees to contain the true state by solving a time-varying linear matrixinequality. Simulation results are provided to demonstrate the effectiveness of the proposed method.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Most filtering approaches require system noises includingprocess noise and measurement noise in a stochastic frameworkand then provide a probabilistic state estimation (Anderson &Moore, 1979; Yang & Hung, 2002; Yang, Wang, & Hung, 2002;Yang, Wang, Ho, & Liu, 2006). The probabilistic nature of theestimates leads to the use of mean and variance to describe thestate spreads (distributions). These spreads cannot guarantee thatthe state is included in some region, because they are not hardbounds. However, in many real-world applications, such as, targettracking and attack, system guidance and navigation, they need100% confidence to be estimated. This has motivated to develop anellipsoidal state estimation (Kurzhanski & Valyi, 1996). The idea ofthe ellipsoidal state estimation is to provide a set of state estimatesin state space which always contains the true state of the systemby assuming hard bounds instead of stochastic descriptions on

I This paper was not presented at any IFAC meeting. This work has beensupported in part by the Engineering and Physical Sciences Research Council(EPSRC) of the U.K. under Grant EP/C007654/1, the National Nature ScienceFoundation of China under Grant 60874059 and 60604027. This paper wasrecommended for publication in revised form by Associate Editor Andrey V. Savkinunder the direction of Editor Ian R. Petersen.∗ Corresponding address: East China University of Science and Technology, 130Meilong Road, 200237 Shanghai, China. Tel.: +86 0 21 64251605; fax: +86 0 2164253325.E-mail addresses: Fuwen.Yang@brunel.ac.uk, fwyang@ecust.edu.cn (F. Yang),

Yongmin.Li@brunel.ac.uk (Y. Li).

0005-1098/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2009.04.011

the system noises (Bertsekas & Rhodes, 1971; Schweppe, 1968).The actual estimate is a set in state space rather than a singlevector. These methods are therefore known as set-membership orset-valued state estimation (filtering) (Bertsekas & Rhodes, 1971;Morrell & Stirling, 1991; Schweppe, 1968; Witsenhausen, 1968).Weprefer to adopt thename set-membership filtering in this paperas it is easy to distinguish between a set estimation and a pointestimate in the stochastic framework.Set-membership filtering problem was first considered by

Witsenhausen (1968). The set of all possible values of the statescompatible with the measurement of outputs is completely char-acterised by their support functions. An ellipsoidal approximationalgorithm was provided by Schweppe (1968). In this algorithm,the measurements are used to calculate recursively a boundingellipsoid to the set of possible states, under the assumption thatthe set containing the initial condition and the process and mea-surement noises is, or can be approximated by ellipsoids. Thesolution to a set-membership filtering problem with the instanta-neous constraints was determined by using the results derived foran energy constraint in Bertsekas and Rhodes (1971). The result-ing estimator is similar to that proposed by Schweppe (1968), butit has an important advantage that the gainmatrix does not dependon the particular output measurements and is therefore precom-putable. Recently, many researchers have attempted to deal withthe set-membership filtering problems with various methods. Forexample, a convex optimisation approachwas applied to deal withthe robust set-membership filtering for the systems with norm-bounded uncertainty in the system matrices (Ghaoui & Calafiore,2001). An ellipsoidal state boundingmethodwas developed to pro-vide an optimal outer approximation of the sum and intersection

F. Yang, Y. Li / Automatica 45 (2009) 1896–1902 1897

of ellipsoids for set-membership filtering in Durieu, Walter, andPolyak (2001);Maskarov andNorton (1996); Polyak, Nazin, Durieu,and Walter (2002). A recursive scheme for constructing an ellip-soidal state estimation set of all states consistent with the mea-sured output and the given noise and unstructured uncertaintydescribed by a sum quadratic constraint was presented in Petersenand Savkin (1999); Ra, Jin, and Park (2004); Savkin and Petersen(1995, 1998). In the existing literature concerning set-membershipfiltering techniques, it is implicitly assumed that the measure-ments are working under the linear condition such that the effectof possible amplitude saturation is ignored (Ghaoui & Calafiore,2001; Polyak et al., 2002; Ra et al., 2004; Savkin & Petersen, 1995,1998). In practical applications, however, sensor saturation arisesfrequently in measurements due to physical limitations on sen-sors (Sadhar & Rajagopalan, 2004). This motivates us to investigatethe set-membership filtering problem with sensor saturation.Analysis and design for systems with sensor saturation have

been addressed by several authors (Cao, Lin, & Chen, 2003; Hu &Lin, 2001; Koplon, Hantus, & Sontag, 1994; Kreisselmeier, 1996;Lin & Hu, 2001; Xiao, Cao, & Lin, 2004; Zuo, Wang, & Huang,2005). For example, the stabilization for linear systems subject tosensor saturation has been considered in Kreisselmeier (1996); Linand Hu (2001). An output feedback control with sensor saturationhas been proposed for linear continuous-time systems in Caoet al. (2003). An extension to linear discrete-time systems hasbeen presented in Zuo et al. (2005). The observability of linearsystemswith saturated outputs has been discussed in Koplon et al.(1994). A robust filter has been designed for discrete-time systemswith sensor saturation and applied to the digital transmultiplexersystems in Xiao et al. (2004). In this paper, we consider the set-membership filtering problem for discrete-time systems subjectto sensor saturation. Our goal is to provide a region of stateestimate where the true state resides despite the presence of bothmeasurement saturation, and unknown-but-bounded process andmeasurement noises. For this purpose, a sufficient condition forthe existence of set-membership filter is derived for discrete-timesystems with sensor saturation. A convex optimisation methodis proposed to determine a state estimation ellipsoid that is a setof states compatible with sensor saturation and unknown-but-bounded process and measurement noises. A recursive algorithmis developed for computing the ellipsoid that guarantees to containthe true state by solving a time-varying linear matrix inequality.The rest of this paper is organised as follows. The set-

membership filter design problem is formulated in Section 2for discrete time-varying systems subject to sensor saturation. Aconvex optimisation method for computing the state estimationellipsoid is developed in Section 3. Section 4provides an illustrativeexample to demonstrate the effectiveness of our algorithm.Conclusions and future directions are described in Section 5.

Notation. The notation X ≥ Y (respectively, X > Y ) where X and Yare symmetric matrices, means that X−Y is positive semi-definite(respectively, positive definite). The superscript T stands formatrixtranspose. The notation trace(P) denotes the trace of P .

2. Problem formulation

Consider a discrete time-varying system in the presence ofsensor saturation

xk+1 = Akxk + Fkuk + Bkwk, (1)

yk = σ(Ckxk)+ Dkvk, (2)

where xk ∈ Rn is the system state; uk ∈ Rl is the known deter-ministic input; yk ∈ Rm is the measurement output; Ak, Bk, Ck,

Dk and Fk are known time-varying matrices with appropriatedimensions; The saturation function σ(·):Rm 7→ Rm is defined as

σ(r) =

σ1(r1)σ2(r2)· · ·

σm(rm)

(3)

with σi(ri) = sign(ri)min{ri,max, |ri|}, where ri,max denotes the ithelement of the vector rmax, the saturation level; wk ∈ Rr is theprocess noise and vk ∈ Rp is the measurement noise, which isassumed to be confined to specified ellipsoidal sets:

Wk := {wk : wTkQ−1k wk ≤ 1}, (4)

Vk := {vk : vTkR−1k vk ≤ 1}, (5)

where Qk = Q Tk > 0 and Rk = RTk > 0 are known matriceswith compatible dimensions; the initial state x0 belongs to a givenellipsoid:

X0 := {x0 : (x0 − x0)TP−10 (x0 − x0) ≤ 1}, (6)

where x0 is an estimate of x0 which is assumed to be given, andP0 = PT0 > 0 is a known matrix.

Remark 1. Sensor saturation exists in many practical applicationsdue to the use of cheaper sensors with inadequate range ininstrument device, amplifier saturation of electronic circuit, ratelimitation in mechanical systems. The examples are flight controlsystems (Cao et al., 2003) and restoration of images (Sadhar& Rajagopalan, 2004). Here we only consider the Occasionalmeasurement saturations. Deep saturations are beyond our studyin this paper.

In this paper, a filter based on the current measurement isconsidered for the system (1)–(2) subject to the saturation (3),which is of the form:

xk+1 = Gkxk + Fkuk + Lkyk+1, (7)

where xk ∈ Rn is the state estimate of xk. Gk and Lk are the filterparameters to be determined.Our objective is to determine an ellipsoid

Xk+1 := {xk+1 : (xk+1 − xk+1)TP−1k+1(xk+1 − xk+1) ≤ 1}, (8)

for the state xk+1, given the measurement information yk+1 at thetime instant k+1 for the process noisewk ∈ Wk and the measure-ment noise vk ∈ Vk subject to the saturation (3). In other words,we look for Pk+1 and xk+1 such that

(xk+1 − xk+1)TP−1k+1(xk+1 − xk+1) ≤ 1, (9)

subject towk ∈ Wk, vk ∈ Vk and (3).This problem will be referred to as a set-membership filter

design problem.

3. Set-membership filter design with sensor saturation

In this section, we will design a set-membership filter for thesystem (1)–(2) subject to sensor saturation. We start with somebasic notions that we are interested in. We then recall two usefullemmas for our following development. After that we derive theexistence conditions for the set-membership filter that the truestate is guaranteed to reside in a set of state estimates. We finallyprovide the convex optimisation recursive algorithm for designingthe set-membership filter.

1898 F. Yang, Y. Li / Automatica 45 (2009) 1896–1902

σ(Ckxk)H2Ckxk

H1Ckxk

Ckxksmax–smax

–ri,max

ri,max

Fig. 1. The saturation type nonlinearity.

Definition 1 (Cao et al., 2003; Khalil, 1996). A nonlinearity ψ: Rm7→ Rm is said to satisfy a sector condition if

(ψ(v)− K1v)T(ψ(v)− K2v) ≤ 0, ∀v ∈ Rm (10)

for some real matrices K1, K2 ∈ Rm×m, where K = K2 − K1 isa positive-definite symmetric matrix. In this case, we say that ψbelongs to the sector [K1, K2].

If we assume that there exist the diagonal matrices H1 and H2such that 0 ≤ H1 < I ≤ H2, then the saturation function σ(Ckxk)in (2) can be written as follows

σ(Ckxk) = H1Ckxk + ψ(Ckxk), (11)

where ψ(Ckxk) is a nonlinear vector-valued function whichsatisfies a sector condition with K1 = 0 and K2 = H , in whichH = H2 − H1, i.e., ψ(Ckxk) satisfies the following inequality:

ψT(Ckxk)(ψ(Ckxk)− HCkxk) ≤ 0. (12)

Remark 2. In theory, we should choose H1 as zero to guaranteethat the inequality (12) holds in whole. However, in practicalapplications, Ckxk is limited, i.e., Ckxk is not greater than smax inFig. 1. So we can use a smaller sector (a bigger H1) to cover thesaturation type nonlinearities and guarantee that the inequality(12) is satisfied. If H1 is chosen too small, the saturation functionwill belong to a bigger sector (see Fig. 1), and the estimated regionwill become bigger too.

In the following set-membership filter design for sensorsaturation, we will take into account the saturation constraints(11) and (12). Before providing the solution to the set-membershipfiltering problem with sensor saturation, we recall the followingtwo useful lemmas:

Lemma 1 (S-Procedure Boyd, Ghaoui, Feron, & Balakrishnan, 1994;Skelton, Iwasaki, & Grigoriadis, 1998). Let Y0(η), Y1(η), . . ., Yp(η) bequadratic functions of η ∈ Rn

Yi(η) = ηTTiη, i = 0, 1, . . . , p, (13)

with Ti = T Ti . Then, the implication

Y1(η) ≤ 0, . . . , Yp(η) ≤ 0 H⇒ Y0(η) ≤ 0, (14)

holds if there exist τ1, . . . , τp > 0 such that

T0 −p∑i=1

τiTi ≤ 0. (15)

Lemma 2 (Schur Complements Boyd et al., 1994). Given constantmatrices L1, L2, L3 where L1 = LT1 and L2 = L

T2 < 0, then

L1 − LT3L−12 L3 ≤ 0,

if and only if[L1 LT3L3 L2

]≤ 0,

or equivalently[L2 L3LT3 L1

]≤ 0.

We are now in position to provide the main results in thefollowing theorem. The existence conditions are developed for theset-membership filter that the true state is guaranteed to reside ina set of state estimates.

Theorem 1. For the system (1)–(2) subject to the sensor saturation(3), suppose that the state xk belongs to its state estimation ellipsoid(xk − xk)TP−1k (xk − xk) ≤ 1. Then one-step ahead state xk+1 residesin its state estimation ellipsoid (xk+1− xk+1)TP−1k+1(xk+1− xk+1) ≤ 1,if there exist Pk+1 > 0, Gk, Lk, τ1 > 0, τ2 > 0, τ3 > 0, τ4 > 0 suchthat[−Pk+1 Π(xk, uk)

ΠT(xk, uk) −Θ(τ1, τ2, τ3, τ4)

]≤ 0. (16)

Moreover the centre of the state estimate ellipsoid is determined by

xk+1 = Gkxk + Fkuk + Lkyk+1, (17)

where Θ(τ1, τ2, τ3, τ4) and Π(xk, uk) are given in Box I, and Φk isgiven in Box II.

Proof. For simplicity, we denote ψ(Ckxk) by ψk, then one-stepahead estimation error xk+1 − xk+1 is written as:

xk+1 − xk+1= Akxk + Fkuk + Bkwk − Gkxk − Fkuk − Lkyk+1= Akxk + Bkwk − Gkxk − Lk[σ(Ck+1xk+1)+ Dk+1vk+1]= Akxk + Bkwk − Gkxk − Lk(H1Ck+1xk+1 + ψk+1 + Dk+1vk+1)= (I − LkH1Ck+1)Akxk − Gkxk − LkH1Ck+1Fkuk+ (I − LkH1Ck+1)Bkwk − Lkψk+1 − LkDk+1vk+1. (20)

On the other hand, if (xk − xk)TP−1k (xk − xk) ≤ 1, then thereexists a z with ‖z‖ ≤ 1 such that (Ghaoui & Calafiore, 2001)

xk = xk + Ekz, (21)

where Ek is a factorisation of Pk = EkETk .By using (21), (20) can be rewritten as

xk+1 − xk+1 = (I − LkH1Ck+1)Akxk − Gkxk − LkH1Ck+1Fkuk+ (I − LkH1Ck+1)AkEkz + (I − LkH1Ck+1)Bkwk− Lkψk+1 − LkDk+1vk+1. (22)

Denoting

η =

1zwkvk+1ψk+1

(23)

and Π(xk, uk), given in Box III, we obtain a compact form for (21)as follows:

xk+1 − xk+1 = Π(xk, uk)η, (24)

F. Yang, Y. Li / Automatica 45 (2009) 1896–1902 1899

8)

9)

Θ(τ1, τ2, τ3, τ4) = τ1Φk + diag(1− τ2 − τ3 − τ4, τ2I, τ3Q−1k , τ4R−1k+1, 0), (1

Π(xk, uk) =[(I − LkH1Ck+1)Akxk − Gkxk − LkH1Ck+1Fkuk (I − LkH1Ck+1)AkEk (I − LkH1Ck+1)Bk −LkDk+1 −Lk

](1

Box I.

Φk =12

0 0 0 0 −(HCk+1Akxk + HCk+1Fkuk)T

0 0 0 0 −(HCk+1AkEk)T

0 0 0 0 −(HCk+1Bk)T

0 0 0 0 0−HCk+1Akxk − HCk+1Fkuk −HCk+1AkEk −HCk+1Bk 0 2

Box II.

Π(xk, uk) =[(I − LkH1Ck+1)Akxk − Gkxk − LkH1Ck+1Fkuk (I − LkH1Ck+1)AkEk (I − LkH1Ck+1)Bk −LkDk+1 −Lk

]Box III.

Hence, (xk+1 − xk+1)TP−1k+1(xk+1 − xk+1) ≤ 1 can be written as

ηT[ΠT(xk, uk)P−1k+1Π(xk, uk)− diag(1, 0, 0, 0, 0)]η ≤ 0. (25)

From (12), we have

ψTk+1(ψk+1 − HCk+1xk+1) ≤ 0. (26)

Substituting (1) and (21) into (26) yields

ψTk+1(ψk+1 − HCk+1Akxk−HCk+1Fkuk − HCk+1AkEkz − HCk+1Bkwk) ≤ 0 (27)

which can be written in η as

ηTΦkη ≤ 0, (28)

whereΦk is as given in Box IV.From (4), (5) and (21), z, wk and vk+1 have the following

constraints‖z‖ ≤ 1,wTkQ

−1k wk ≤ 1,

vTk+1R−1k+1vk+1 ≤ 1.

(29)

which can be expressed in η as:ηTdiag(−1, I, 0, 0, 0)η ≤ 0,ηTdiag(−1, 0,Q−1k , 0, 0), η ≤ 0,ηTdiag(−1, 0, 0, R−1k+1, 0)η ≤ 0.

(30)

Therefore we need to find a condition tomake (25) hold subjectto the saturation constraints (28), and constraints (30). By usingthe S-procedure (Lemma 1), the sufficient condition such that theinequalities (28) and (30) imply (25) to hold is that there existpositive scalars τ1, τ2, τ3, and τ4 such that

ΠT(xk, uk)P−1k+1Π(xk, uk)− diag(1, 0, 0, 0, 0)− τ1Φk

− τ2diag(−1, I, 0, 0, 0)− τ3diag(−1, 0,Q−1k , 0, 0)

− τ4diag(−1, 0, 0, R−1k+1, 0) ≤ 0. (31)

(31) is written in the following compact form:

ΠT(xk, uk)P−1k+1Π(xk, uk)− τ1Φk

− diag(1− τ2 − τ3 − τ4, τ2I, τ3Q−1k , τ4R−1k+1, 0) ≤ 0. (32)

By denoting

Θ(τ1, τ2, τ3, τ4) = τ1Φk

+ diag(1− τ2 − τ3 − τ4, τ2I, τ3Q−1k , τ4R−1k+1, 0), (33)

(32) is written as

ΠT(xk, uk)P−1k+1Π(xk, uk)−Θ(τ1, τ2, τ3, τ4) ≤ 0. (34)

By using the Schur complements (Lemma 2), (34) is equivalentto:[−Pk+1 Π(xk, uk)

ΠT(xk, uk) −Θ(τ1, τ2, τ3, τ4)

]≤ 0. (35)

Thus, if there exist the filter parameters Gk and Lk, and scalarsτ1 > 0, τ2 > 0, τ3 > 0, τ4 > 0 such that (16) holds, then one-stepahead state xk+1 resides in its state estimation ellipsoid (xk+1 −xk+1)TP−1k+1(xk+1 − xk+1) ≤ 1. �

Theorem 1 outlines the principle of determining the currentstate estimation ellipsoid containing xk+1. However, it does notprovide an optimal (minimal) state estimation ellipsoid. Next, weapply the convex optimisation approach (Nesterov & Nemirovski,1994; Vandenberghe & Boyd, 1996) to determine an optimalellipsoid. Pk+1 is obtained by solving the following optimisationproblem:

minPk+1>0,Gk,Lk,τ1>0,τ2>0,τ3>0,τ4>0

trace(Pk+1)

subject to (16)(36)

Remark 3. We can see from Theorem 1 that the inequality (16)is linear to the variables Pk+1, Gk, Lk, τ1, τ2, τ3, τ4. Hence, theoptimisation problem (36) subject to (16) can be solved bythe existing semi-definite programming (SDP) via interior-pointapproach (Nesterov & Nemirovski, 1994; Vandenberghe & Boyd,1996).

Remark 4. The sufficient conditions are provided to estimate astate ellipsoid. In order to reduce the conservativeness, the convexoptimisation method has been proposed in (36). The trace of Pk+1is optimised at each time step in an effort to find the smallestellipsoid for the state estimate. Other measures of the ellipsoid canalso be introduced, for example, determinant (Durieu et al., 2001;Ghaoui & Calafiore, 2001).

Remark 5. It is worthmentioning that a set-value state estimationproblem has been proposed in Petersen and Savkin (1999); Savkinand Petersen (1995, 1998) by designing a robust Kalman filter andthen deriving the bound of (xk+1−xk+1)TP−1k+1(xk+1−xk+1) via usingan analytical method. In this paper, we assume a general structureof the filter as the centre of the state estimate ellipsoid, and then

1900 F. Yang, Y. Li / Automatica 45 (2009) 1896–1902

Φk =12

0 0 0 0 −(HCk+1Akxk + HCk+1Fkuk)T

0 0 0 0 −(HCk+1AkEk)T

0 0 0 0 −(HCk+1Bk)T

0 0 0 0 0−HCk+1Akxk − HCk+1Fkuk −HCk+1AkEk −HCk+1Bk 0 2

Box IV.

use a numerical optimisation method to find a minimum positive-definite matrix Pk+1 such that (xk+1− xk+1)TP−1k+1(xk+1− xk+1) < 1in the sense of trace performance. The advantage of our methodis that it is easy to deal with saturation nonlinearity by usingthe S-procedure. Moreover, the process and measurement noiseswe consider are assumed to be confined to specified ellipsoidalsets, whereas the process and measurement noises consideredin Petersen and Savkin (1999); Savkin and Petersen (1995, 1998)are assumed to satisfy an integral quadratic constraint.

4. An illustrative example

In this section, a simple example is provided to test ouralgorithm. We consider a maneuvering target tracking system andadopt a constant-accelerationmotionmodel as follows (Li & Jilkov,2003)

xk+1 =

1 T T 2/20 1 T0 0 1

xk + [211

]wk, (37)

where T is the sample period, the state xk = [ s(k) s(k) s(k) ]Tis the position, velocity and acceleration of the target at time kT ,respectively.Since the position is measured by radar, the measurement is

described as

yk = σ([1 0 0

]xk)+ vk, (38)

where σ(·) is a saturation function. The saturation function σ(·)can be described as follows:{σ(sk) = sk, if − ri,max ≤ sk ≤ ri,max;σ(sk) = ri,max, if sk > ri,max;σ(sk) = −ri,max, if sk < −ri,max

where sk is the position value of target. When sk is less than thesaturation value r1,max, the sensorwill display the value of sk.Whensk exceeds the saturation value r1,max, the sensor will display thesaturation value r1,max. In this example the saturation value r1,maxis 1.8.In the simulation, the sample period T is chosen as 0.1 second,

and wk and vk as 0.5 sin(2k) and 0.2 sin(20k), respectively. H andH1 are set as H = 0.3 and H1 = 0.7. The initial state is set as x0 =[0 0 0]T, which belongs to the ellipsoid (x0−x0)TP−10 (x0−x0) ≤ 1,

where x0 = [1 1 1]T and P0 =[50 0 00 50 00 0 50

]; Qk = 0.3, and

Rk = 0.1.The simulation results are obtained by solving the convex

optimisation problem (36) subject to (16) under Matlab 6.5 withYALMIP 3.0 and SeDuMi 1.1 (Löfberg, 2004). Fig. 2 shows thatthe true position always resides between the upper bound andlower bound. The upper bound and lower bound are determinedby calculating xk = xk+ Ekz. In this example, we choose z as a unitball. FromFig. 2,we can see that the target belongs to the estimatedregion. The target can be fully tracked. Therefore, our proposedfiltering algorithm is useful for target tracking and target attack.Figs. 3 and 4 also indicate that the true velocity and accelerationreside between their upper bounds and lower bounds, i.e., the

Fig. 2. The true position value, position estimation, its upper bound and lowerbound with the proposed set-membership filtering method.

Fig. 3. The true velocity value, velocity estimation, its upper bound and lowerbound with the proposed set-membership filtering method.

estimated ellipsoids always contain the true states. When thesaturation value is changed to r1,max = 1.6, the simulation resultsshow that the state estimate values are little far away from thetrue state values. Hence, the saturations will make the estimationaccuracy become poor. Moreover, the different values of H1 willget different results. The bigger H1 which cannot guarantee thatthe inequality (12) is satisfied will not give any solution.Nowwe apply the traditional set-membership filteringmethod,

which does not consider the sensor saturation in design, to conductthe simulation. The results are displayed in Figs. 5–7. We can seethe true position value becomes greater than its upper bound atk = 47. This is because the sensor saturation is not taken into ac-count in the filter design. Therefore, our filtering algorithmcandealwith the sensor saturation in measurements. However, it is neces-sary to point out that the traditionalmethod cannot guarantee thatthe true state resides in the state estimation ellipsoid. It does notmean that the target always goes out of the state estimation ellip-soid.

F. Yang, Y. Li / Automatica 45 (2009) 1896–1902 1901

Fig. 4. The true acceleration value, acceleration estimation, its upper bound andlower bound with the proposed set-membership filtering method.

Fig. 5. The true position value, position estimation, its upper bound and lowerbound with the traditional set-membership filtering method.

Fig. 6. The true velocity value, velocity estimation, its upper bound and lowerbound with the traditional set-membership filtering method.

5. Conclusions

This paper has provided a new approach that is able todeal with sensor saturation and unknown-but-bounded processand measurement noises in filtering problem for discrete time-varying systems. A sufficient condition for the existence of

Fig. 7. The true acceleration value, acceleration estimation, its upper bound andlower bound with the traditional set-membership filtering method.

set-membership filter has been derived. A convex optimisationmethod has been proposed to determine a state estimationellipsoid that is a set of states compatible with sensor saturationand unknown-but-bounded process and measurement noises. Arecursive algorithm has been derived for computing the stateestimate ellipsoid which always contains the state. A smallestpossible estimate set is computed recursively by solving thesemi-definite programming problems. An illustrative example hasdemonstrated the feasibility of the proposed filtering methods.The algorithm is computationally attractive for on-line systemswith sensor saturation in the presence of unknown-but-boundedprocess and measurement noises. The much more challengingfuture research topics should be the study of the convergence ofthe algorithms and how to reduce the conservatism of the possibleestimation sets.

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Fuwen Yang received the B.Sc. degree in electrical engi-neering from Anhui University of Science and Technology,China, and theM.Sc. and Ph.D. degrees in control engineer-ing from Huazhong University of Science and Technology,China.He is currently a professor in the School of Information

Science and Engineering at East China University of Sci-ence and Technology, Shanghai, China. Before joining EastChina University of Science and Technology, he workedat Fuzhou University, China; King’s College London, UK;Brunel University, UK; the University of Manchester, UK,

and the University of Hong Kong. He has published more than 100 journal andconference papers in systems control and signal processing. His research interestsinclude H-infinity control and filtering, networked control, fault detection and di-agnosis, signal processing, industrial real-time control, and power electronics.Dr. Yang was a recipient of the Teaching Excellence Award for Young Teachers

in 1995 from Fok Ying Tung Education Foundation, China; four Science and Tech-nology Development Awards in 1996, 1999, 2002 and 2006, from Fujian Provinceof China; and two Youth Science and Technology Awards both in 1998 from FujianProvince of China and from Yun Sheng Foundation. He was listed in Ten Outstand-ing Youth from Fujian Province of China in 1999 and was honored the May 4 YouthMedal from Fujian Province of China in 2000. Since 2001, he has been acting as aState Consultant receiving Special Allowance from the State Council of the People’sRepublic of China.

Yongmin Li received his B.Eng. and M.Eng. in controlengineering from Tsinghua University of China in 1990and 1992 respectively, and his Ph.D. in computer visionfrom Queen Mary, University of London in 2001. Heis currently a faculty member in the Department ofInformation Systems and Computing at Brunel University,UK. Before joining Brunel University, he worked as aresearch scientist in the British Telecom Laboratories. Hisresearch interest covers the areas of automatic control,machine learning, pattern recognition, computer vision,image processing, video analysis, medical imaging and

bio-imaging.Dr. Li received the BMVA Best Scientific Award in 2001 and the Best Paper Prize

in IEEE RATFG 2001. Dr. Li is a senior member of IEEE.