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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI10.1109/TSP.2015.2465305, IEEE Transactions on Signal Processing

1

Robust Scheduling Filter Design for a Class ofNonlinear Stochastic Poisson Signal Systems

Bor-Sen Chen†, Life Fellow, IEEE and Chien-Feng Wu

Abstract—Nonlinear dynamic systems may suffer from bothcontinuous Wiener noise and discontinuous Poisson noise. Thispaper studies a robust filter design for a class of nonlinearstochastic Poisson signal systems with external disturbances.Currently, there are no good filtering design methods to treatthe discontinuous Poisson noise filter problem. Based on the Ito-Levy formula, a robust H∞ filter design is proposed for nonlinearstochastic Poisson signal systems by solving a Hamilton-Jacobi in-equality (HJI) for the robust filter design of a nonlinear stochasticPoisson signal system. However, such an HJI is difficult to solve.Hence, this study employs the Polytopic Linear Model(PLM)scheduling scheme to approximate this HJI by a set of linearmatrix inequalities(LMIs) so that the H∞ robust filter designproblem for a nonlinear stochastic Poisson signal system canbe simplified. The optimal H∞ robust scheduling filter designproblem for the nonlinear stochastic Poisson signal system is alsodiscussed. Since the PLM interpolation method transforms theHJI-constrained optimization problem into an LMI-constrainedoptimization problem which can be efficiently solved using theLMI toolbox in MATLAB, an optimal filtering level γ∗ (theminimum value of the filtering error-to-noise ratio in a meansquare sense) can be achieved. Finally, a simulation example of arobust trajectory estimation problem in an anti-tactical ballisticmissile radar system with discontinuous random maneuveringjets is given to illustrate the design procedure and to confirm theestimation performance of the proposed H∞ robust schedulingfilter.

Index terms— Robust scheduling filter, Polytopic LinearModel (PLM) scheduling, Poisson jump system, Hamilton-Jacobi inequality (HJI), linear matrix inequality (LMI).

I. INTRODUCTION

The H∞ robust filtering problem aims to design a fil-ter via output measurement to estimate the unknown statecombination which guarantees the L2 gain from the externaldisturbance to an estimation error lower than a prescribedfiltering level [1]-[5]. Since H∞ state estimation is usefulfor robust filter design when the transmitted signals to bedetected are embedded in state variables and corrupted bynoises or disturbances in the transmission process, robuststate estimation [1]-[22] has become a significant topic inrobust filter design and robust state feedback control design.In contrast with the well-known H2 Kalman filter [23], a keyadvantage of the robust filter is that it is not necessary toexactly know the statistical properties of external disturbances;the filter can be applied by assuming the external disturbances

This manuscript submitted at Apr 7, 2015; revised June 1, 2015; acceptedJuly 24,2015.

The authors are with the Department of Electrical Engineering, NationalTsing Hua University, Hsinchu,Taiwan 30013.(e-mail:[email protected];[email protected])

to have bounded energy. Some practical applications of H∞filtering in signal processing can be found in [8], [14]-[15].

The H∞ filter and controller design problems of a stochasticlinear system with Wiener noise have been a popular researchtopic [3], [4], [7], [16]-[22]. Over the past 20 years , stochasticsystems with Poisson signal noise [24]-[27] has also receivedincreasing focus in the literature. Since many stochastic sys-tems in engineering, economics and biology have both contin-uous Wiener and discontinuous Poisson noises [16], [20], [27],it is appealing to extend robust control design to robust filterdesign for stochastic systems with both continuous Wienerand discontinuous Poisson noise. However, the robust filterdesign for a nonlinear stochastic Poisson process still needsmore attention for this to be appropriate. Poisson noise canbe captured as event-driven uncertainties, such as corporatedefaults, operational failures, hostile interferences, or insuredevents [28]-[32]. Wiener noise can be regarded as continuousrandom fluctuations, such as thermal fluctuation [33]. Indeed,such Wiener and Poisson noises are found in a considerablevariety of applications, including physical sciences, biology,engineering, signal processing and stochastic resonance. Infinancial and actuarial modeling and other areas of application,such Poisson signals are often used to describe the dynamics ofvarious state variables with discontinuous jumps. This Poissonsignal may represent, for instance, asset prices, credit ratings,stock indices, interest rates, exchange rates or commodityprices [28]-[30]. These complex systems are often modeledby nonlinear stochastic Poisson signal systems. Therefore, theestimation of the desired state variables of these nonlinearstochastic Poisson signal systems is an important topic in thefields of filtering and control. Since a nonlinear stochasticPoisson signal system consists of continuous Wiener noise(Brownian motion) W(t) and discontinuous Poisson noiseN (t), the robust H∞ filter design is more difficult to deal withthan the conventional H∞ filter design with only continuousnoise. Extended Kalman filtering algorithms have been widelyapplied to treat filter design problems in nonlinear stochasticsystems. However, the filters of nonlinear stochastic Poissonsystems cannot be designed by extended Kalman filteringalgorithms because the latter are based on sampled (discrete)nonlinear stochastic systems and the Poisson jumping processmay lose information in the sampling process. Currently, thereexists no efficient method to solve these robust filter designproblems of nonlinear stochastic Poisson signal systems dueto the nonlinear discontinuity of corrupted Poisson noises.

Based on the Ito-Levy formula [24], [25], we must solvea second-order HJI and second-order HJI-constrained opti-mization problem for the robust H∞ filtering design and the



2

optimal robust H∞ filter design of the nonlinear stochasticPoisson signal system, respectively. However, solving theseproblems for the robust H∞ filter design of nonlinear stochas-tic Poisson signal system remains particularly difficult. Thus,Polytopic Linear Models (PLM) [35] are proposed to inter-polate several local linear stochastic Poisson signal systemsto approximate the given nonlinear stochastic Poisson signalsystem; then, its corresponding HJI can be also interpolated bya set of linear matrix inequalities (LMIs). In this situation, anLMI-constrained optimization problem can replace the HJI-constrained optimization problem of the robust filter designfor nonlinear stochastic Poisson systems and can be solvedefficiently using a MATLAB Toolbox [36]. Moreover, sincethe approximation errors are also considered in the designprocedure, the proposed robust scheduling filter can efficientlyestimate the unknown system state and simultaneously guar-antee a prescribed H∞ filtering performance, i.e. the externalnoise to the estimation error (the error-to-noise ratio in meansquare sense) is lower than a prescribed filtering level.

The main contributions of this paper are: (i) The first inves-tigation of the robust H∞ filter design problem for nonlinearstochastic Poisson signal system with continuous Wiener noiseand discontinuous Poisson noise, and (ii) The proposal ofthe PLM scheduling interpolation method for scheduling theHJIs using a set of LMIs to simplify the design procedureof the H∞ robust filter for the nonlinear stochastic systemswith Poisson and Wiener noises using the LMI toolbox inMATLAB.

The remainder of this paper is organized as follows: theH∞ filter settings for nonlinear stochastic Poisson signalsystems are given in Section II. In Section III, the robustH∞ scheduling filter design for a nonlinear stochastic Poissonsignal system is introduced. Section IV provides the simulationexample of a robust trajectory estimation of reentry vehiclesby radar to illustrate the design procedure and confirm theperformance of the proposed filter design. Finally, conclusionsare given in Section V.

Notation:For the convenience of design, we adopt the following

notations throughout this paper: R+ , [0,∞); In , n-by-n identity matrix; AT : the transpose of matrix A; A ≥0(A > 0):symmetric positive semi-definite (symmetric posi-tive definite) matrix A; I: identity matrix; C2 (Rnx): class offunctions V (·) twice continuously differential with respect tox(t) ; Vx: The gradient column vector of nx-dimensional twicecontinuously differentiable function V (x(t))(, i.e. (∂V (x(t)))

∂x(t) );Vxx: The Hessian matrix with elements of second partialderivatives of nx-dimensional twice continuously differen-tiable function V (x(t))(, i.e. (∂

2V (x(t)))∂x2(t) ); N∅ : the tatality of

P-null sets for the given complete probability space (Ω,F ,P);L2F (R+,Rny ): the space of nonanticipative stochastic pro-

cesses y ∈ Rny relative to an increasing family Ftt∈R+

of σ-algebra Ft ⊂ F satisfying ‖y(t)‖L2F (R+,Rny ) ,(

E∫∞0‖y(t)‖2dt

) 12 < ∞; E·: the expectation opera-

tor; P·: the probability measure function; diag(a, b) ,[a 00 b

];

[M1 M2

MT2 M3

],

[M1 ?MT

2 M3

].

II. THE H∞ FILTER SETTINGS FOR NONLINEARSTOCHASTIC POISSON SIGNAL SYSTEMS

Let (Ω,F ,P) be a complete probability space with filtra-tion Ftt≥0 = σ N (s) : s ≤ t ∨ σ W (s) : s ≤ t ∨ N∅generated by the mutually independent Poisson process N (t)and standard Wiener process W (t). Consider the followingnonlinear stochastic Poisson signal system:

dx (t) = [f (x (t)) + g (x (t))v1 (t)] dt+h (x (t)) dW (t) + i (x (t)) dN (t) ,Y (t) = q (x (t)) + k (x (t))v2 (t)+j (x (t))w (t) +m (x (t))n (t) ,

s (t) = Gx (t) ,f (0) = h (0) = i (0) = 0, g (0) = 0q (0) = j (0) = m (0) = 0, and k (0) = 0

(1)

where x (t) ∈ L2F (R+,Rnx) is the system state; Y (t) ∈ Rny

is the measurement output; v1 (t) ∈ L2F (R+,Rnv1 ) and

v2 (t) ∈ L2F (R+,Rnv2 ) are used to denote the exogenous

disturbance signals; and s (t) denotes the system state combi-nation used for estimation where G is a combination matrix.If the whole state vector x (t) needs to be estimated fromY (t), then G = I .(That is, if we want to estimate onlythe last state xn (t) of x (t), then G = diag(0, 0, ..., 1)) ThetermsW (t) and N (t) are the one-dimensional (1-D) standardWiener process and Poisson process, respectively. The termsf , g, h, i, q, k, j and m are Borel measurable C1 functionswith appropriate dimension. Moreover, the nonlinear stochasticPoisson signal system in (1) satisfies the Lipschitz and lineargrowth conditions [37] so that it has a unique solution. Thestate-dependent stochastic term h (x (t)) dW (t) can be re-garded as an intrinsic continuous random parameter fluctuationdue to Wiener noise [4], [22]. The measurement noise w (t)denotes the white noise with dW (t) = w (t) dt and n (t) isa discontinuous measurement noise with dN (t) = n (t) dtwhich is caused by the Poisson counting process N (t). Thus,the term i (x (t)) dN (t) can be regarded as the discontinuousinternal jumping process due to the discontinuous Poissonnoise in the given nonlinear stochastic Poisson system.

Some important properties ofW (t) and N (t) in this paperare given as follows [25]:

1) N (0) = 0.2) dN (t) = 1, when t = tjump , else dN (t) = 0, where

tjump is the jumping time.3) E dN (t) = λdt where λ is the Poisson jump intensity.4) E dN (t) dt = 0.5) P N (t) = k = (λt)

kexp (−λt) /k!, where ∀k ∈ N+,

λ > 0 and t ≥ 0.6) E W (t) = E dW (t) = 0.7) E dW (t) dW (t) = dt.8) E dN (t) dW (t) = 0.For convenience, some properties of nonlinear stochastic

Poisson signal system should be addressed. Since the designeralways needs to transform a filtering design problem to astability problem of the filtering error dynamic system, we firstdiscuss the asymptotic stability problem for the equilibriumpoint xe = 0. If the equilibrium point xe 6= 0, the originshould be shifted to xe for the convenience of analysis.



3

Lemma 1 ([24], [25] ). Let V (·) ∈ C2 (Rnx) and V (·) ≥ 0.For the nonlinear stochastic Poisson signal system in (1), theIto-Levy formula of V (x (t)) is given as follows:

dV (x (t)) =[V Tx f (x (t)) + V Tx g (x (t))v (t)

+ 12h

T (x (t))Vxxh (x (t))]dt+ V Tx h (x (t)) dW (t)

+ V (x (t) + i (x (t)))− V (x (t)) dN (t)(2)

Proposition 1. Consider the following nonlinear stochasticPoisson signal system with the equilibrium point xe = 0

dx (t) = f (x (t)) dt+h (x (t)) dW (t)+i (x (t)) dN (t) (3)

If there exists a Lyapunov function V (·) ∈ C2 (Rnx) satisfyingthe following two inequalities

m1 ‖x (t)‖22 ≤ V (x (t)) ≤ m2 ‖x (t)‖22 , (4)

where m1,m2 > 0, and the HJI in (5)

V Tx f (x (t)) + 12h

T (x (t))Vxxh (x (t))+λ V (x (t) + i (x (t)))− V (x (t)) < 0 ,

(5)

then the nonlinear stochastic system in (3) is asymptoticallystable in probability.

Proof. By applying Lemma 1, we have

E dV (x (t)) = EV Tx f (x (t)) + 1

2hT (x (t))Vxx

h (x (t)) + λ V (x (t) + i (x (t)))− V (x (t)) dt(6)

If the inequality in (5) holds, then we can conclude

E dV (x (t)) ≤ E−m3 ‖x (t)‖22

dt

for some m3 > 0.By using inequalities (4), we get

dE V (x (t))dt

≤ −m3

m2E V (x (t)) (7)

Based on the inequality in (7), we have

E V (x (t)) ≤ E V (x (0)) exp

(−m3t

m2

)(8)

and from (4) it also implies

E‖x (t)‖22

≤ E

V (x (0))

m1

exp

(−m3t

m2

)(9)

so thatlimt→∞

E‖x (t)‖22

= 0 (10)

Since a nonlinear stochastic system is asymptotically stablein mean square implies it is also asymptotically stable inprobability [46], we finish the proof.

For the subsequent analysis, the following well-knownlemma is beneficial for our filter design.

Lemma 2 ([3], [40]). For any matrices (or vectors) M1

and M2 with appropriate dimensions, we have the followinginequality

MT1 M2 +MT

2 M1 ≤MT1 PM1 +MT

2 P−1M2 (11)

where P is any positive-definite symmetric matrix.

Since the robust filtering problem of nonlinear stochasticPoisson signal system is equivalent to the asymptotical sta-bility problem of the filtering error system, the asymptoticalstability and robust H∞ stability of nonlinear stochastic Pois-son signal systems in (1) are discussed first before addressingdesign.

Proposition 2. Suppose xe ≡ 0 is an equilibrium point of thefollowing nonlinear stochastic Poisson signal system

dx (t) = [f (x (t)) + g (x (t))v (t)] dt+h (x (t)) dW (t) + i (x (t)) dN (t)

s (t) = Gx (t)f (0) = h (0) = i (0) = 0 and g (0) = 0

(12)

with the external disturbance v (t) ∈ L2F (R+,Rnv ) and

v (t) 6= 0. If there exists a positive Lyapunov solutionV (·) ∈ C2 (Rnx) with V (x (t)) ≥ 0 and V (0) = 0 forsolving the following HJI

V Tx f (x (t)) + 14γ2V

Tx g (x (t)) gT (x (t))Vx

+xT (t)GTGx (t) + 12h

T (x (t))Vxxh (x (t))+λ V (x (t) + i (x (t)))− V (x (t)) < 0

(13)

then (i) the nonlinear stochastic signal system in (12) isasymptotically stable in probability in the case v (t) = 0, and

(ii) for a given disturbance filtering level γ > 0, thefollowing robust H∞ stability holds:

‖s (t)‖2L2F≤ γ2 ‖v (t)‖2L2

F+ E V (x (0)) for V (x (0)) 6= 0

(14)

‖s (t)‖2L2F≤ γ2 ‖v (t)‖2L2

Ffor V (x (0)) = 0 (15)

Proof. It is clear that

E∫∞

0sT (t) s (t) dt

= E

∫∞0

sT (t) s (t) + dV (x (t)) dt

+E V (x (0)) −E

limt→∞

V (x (t))

(16)By applying (6), we have

E∫∞

0sT (t) s (t) dt

= E

∫∞0

xT (t)GTGx (t) dt

= E∫∞

0xT (t)GTGx (t) dt

+ E V (x (0))

−E

limt→∞

V (x (t))

+ E∫∞

0

(V Tx f (x (t))

+V Tx g (x (t))v (t) + 12h

T (x (t))Vxxh (x (t))+λ [V (x (t) + i (x (t)))− V (x (t))]) dt

(17)By using Lemma 2 with MT

1 =(12

)V Tx g (x (t)), M2 =

v (t) and P = γ2I , the term V Tx g (x (t))v (t) from (17) canbe bounded as follows

V Tx g (x (t))v (t)=(12

)V Tx g (x (t))v (t) +

(12

)vT (t) gT (x (t))Vx

≤ 14γ2V

Tx g (x (t)) gT (x (t))Vx + γ2vT (t)v (t)

(18)

Replacing the term V Tx g (x (t))v (t) in (17) by the inequal-ity in (18), we get the following two inequalities

xT (t)GTGx (t) + V Tx f (x (t)) + 12h

T (x (t))Vxxh (x (t))+V Tx g (x (t))v (t) + λ V (x (t) + i (x (t)))− V (x (t))≤ xT (t)GTGx (t) + V Tx f (x (t)) + γ2vT (t)v (t)+ 1

4γ2VTx g (x (t)) gT (x (t))Vx + 1

2hT (x (t))Vxxh (x (t))

+λ V (x (t) + i (x (t)))− V (x (t))(19)



4

and

E∫∞

0sT (t) s (t) dt

≤ E

∫∞0

xT (t)GTGx (t) dt

+E∫∞

0

(V Tx f (x (t)) + 1

4γ2VTx g (x (t)) gT (x (t))Vx

+ 12h

T (x (t))Vxxh (x (t))dt+ λ [V (x (t) + i (x (t)))

−V (x (t))]) dt+ E∫∞

0γ2vT (t)v (t) dt

+E V (x (0))

(20)If the HJI in (13) is holds, then the inequalities in (14) and

(15) will be held.Based on Proposition 1, if the HJI in (13) holds and v (t) =

0, then the nonlinear stochastic Poisson signal system in (1)is asymptotically stable in probability.

Since the system states are not always completely available,a nonlinear stochastic H∞ filter needs to be constructedto estimate these system states for other applications. Thenonlinear stochastic H∞ filtering problem is to design a robustfilter to estimate the unknown system states in s (t) from theoutput measurement signal Y (t) in (1), which guarantees theL2 gain (from the external noise v (t) to the filtering errore (t)) to be lower than a prescribed filtering level γ > 0.

Now, we construct the following filtering to estimate s (t)of the nonlinear stochastic Poisson signal system in (1)

dx (t) =

(f (x (t)) + L (x (t))

[Y (t)− Y (t)

])dt

Y (t) = q (x (t))s (t) = Gx (t)

(21)where L (x (t)) is defined as the filter gain to be designed toachieve the H∞ robust filtering performance.

For the convenience of filter design, the nonlinear stochasticPoisson signal system in (1) and nonlinear state estimationequation in (21) are combined as an augmented stochasticPoisson signal system.

Set the augmented state x (t) and G as

x (t) =[xT (t) xT (t)

]T(22)

G =[G −G

]. (23)

and let the filtering error e (t) be defined as

e (t) , s (t)− s (t) = Gx (t) (24)

From (1), (21) and (24), we can obtain the followingfiltering error equations based on the augmented stochasticPoisson signal system of (1) and (21):

dx (t) = [fe (x (t)) + ge (x (t))v (t)] dt+he (x (t)) dW (t) + ie (x (t)) dN (t)

e (t) = Gx (t)(25)

where

fe (x (t)) =[fT (x (t)) fT (x (t) , x (t))

]T,

ge (x (t)) =[gT (x (t)) gT (x (t) , x (t))

]T,

v (t) =[vT1 (t) vT2 (t)

]T,

he (x (t)) =[hT (x (t)) hT (x (t) , x (t))

]T,

ie (x (t)) =[iT (x (t)) ıT (x (t) , x (t))

]T,

f (x (t) , x (t)) = f (x (t)) + L (x (t)) [q (x (t))− q (x (t))] ,g (x (t) , x (t)) = (L (x (t)) [k (x (t))− k (x (t))]) ,

h (x (t) , x (t)) = (L (x (t)) [j (x (t))− j (x (t))]) ,andı (x (t) , x (t)) = (L (x (t)) [m (x (t))−m (x (t))])

Thus, the H∞ robust filtering design problem of the nonlinearstochastic Poisson signal system in (1) can be seen to findan estimation gain L (x (t)) such that for a given disturbancefiltering level γ > 0, the following H∞ filtering performanceholds:

‖e (t)‖2L2F≤ γ2 ‖v (t)‖2L2

F+ E V (x (0)) for V (x (0)) 6= 0

(26)

‖e (t)‖2L2F≤ γ2 ‖v (t)‖2L2

Ffor V (x (0)) = 0 (27)

Remark 1. From the above analysis, the H∞ robust filteringdesign problem in (21) of the nonlinear stochastic Poissonsignal system in (1) can be considered as the H∞ robust sta-bilization problem of filtering error dynamic equation in (25).Therefore, the results of asymptotical stability in probability inProposition 1 and H∞ robust stability in Proposition 2 can beemployed to treat the asymptotical filtering (estimation) andH∞ robust filtering design problem, respectively.

Theorem 1. For the H∞ filter design of nonlinear stochas-tic Poisson signal system (1) with a prescribed disturbancefiltering level γ, if the following HJI has a positive solutionV (·) ∈ C2

(R2n

)with V (0) = 0

Ψ (V (x (t))) , Vxfe (x (t)) + 14γ2V

Tx ge (x (t)) gTe (x (t))

Vx + xT (t) GT Gx (t) + 12h

Te (x (t))Vxxhe (x (t))

+λ V (x (t) + ie (x (t)))− V (x (t)) < 0,(28)

then (i) The augmented nonlinear stochastic Poisson signalsystem in (25) is asymptotically filtering, i.e. e (t) → 0 inprobability in the case where v (t) = 0, and (ii) For a givendisturbance filtering level γ > 0, the following H∞ robustfiltering performance holds:

‖e (t)‖2L2F≤ γ2 ‖v (t)‖2L2

F+ E V (x (0)) for V (x (0)) 6= 0

(29)

‖e (t)‖2L2F≤ γ2 ‖v (t)‖2L2

Ffor V (x (0)) = 0 (30)

Proof. Since the filter design problem in nonlinear stochasticPoisson signal system in (1) is equivalent to the stabilizationproblem in the augmented system in (25), the proof can beobtained from Proposition 2 with s (t) and x (t) in (12) being



5

replaced by e (t) = Gx (t) and x (t) =[xT (t) xT (t)

]Tin (25), respectively.

Remark 2. (i) If the nonlinear stochastic system in (1) hasonly Wiener noise, then the HJI in (28) for the H∞ robustfiltering performance in (29) or (30) can be reduced to

Vxfe (x (t)) + 14γ2V

Tx ge (x (t)) gTe (x (t))Vx

+xT (t) GT Gx (t) + 12h

Te (x (t))Vxxhe (x (t)) < 0

(31)

(ii) If the nonlinear stochastic system in (1) has only Poissonnoise, then the HJI in (28) for the H∞ filtering performancein (29) or (30) can be reduced to

Vxfe (x (t)) + 14γ2V

Tx ge (x (t)) gTe (x (t))Vx

+xT (t) GT Gx (t) + λ V (x (t) + ie (x (t)))−V (x (t)) < 0

(32)

Therefore, the nonlinear H∞ robust filter design is tospecify the filter gain L (x (t)) in the nonlinear filter (21)so that the HJI in (28) has a positive energy-like functionsolution V (x (t)) > 0 with V (0) = 0. Then, the optimalrobust H∞ filter design becomes how to solve the followingHJI-constrained optimization problem

γ0 = minL(x(t)) γsubject to V (x (t)) > 0, and HJI in (28) (33)

i.e. to specify the filter gain L (x (t)) in the nonlinear stochas-tic filter system (21) to let filtering level γ in (33) be as smalla value as possible to achieve the optimal filtering level γ0.

III. ROBUST H∞ SCHEDULING FILTER DESIGN FORNONLINEAR STOCHASTIC POISSON SIGNAL SYSTEMS

In general, it is very difficult to directly solve the HJIin (28) for the H∞ filter design of the nonlinear stochasticPoisson signal system in (1). To overcome this issue, wepropose a PLM scheduling scheme based on a regime de-composition of the operating space of a nonlinear stochasticPoisson signal system. Suppose the operating space Φ of thenonlinear Poisson signal system in (1) can be decomposedinto l local compact operation regimes with operating basisvector xi ∈ Φi, i.e. Φi, i = 1.2...l , and Φ ⊂ ∪li=1Φi.Based on the l local operation regimes, we can choose anappropriate local linear model in each local operation regimeso that the nonlinear stochastic Poisson signal system in (1)can be approximated by the PLM scheduling interpolation oflocal linear stochastic systems [35], [37], [41].

Suppose the nonlinear stochastic Poisson signal system in(1) can be linearized at the operation basis vector xi and itslocal operation regimes are given as follows:

∂f(x(t))∂x(t) , ∂g(x(t))

∂x(t) , ∂h(x(t))∂x(t) , ∂i(x(t))

∂x(t) ,∂q(x(t))∂x(t) , ∂k(x(t))∂x(t) ,

∂j(x(t))∂x(t) , ∂m(x(t))

∂x(t) | x (t) = xi ∈ Φi ⊂ Φ

= Ai, Bi, Ci, Di, Qi, Ki, Ji, Mi

where Ai, Bi, Ci, Di, Qi, Ki, Ji, and Mi are constantmatrices with appropriate dimensions for i = 1, 2, ..., l. Thenthe nonlinear stochastic Poisson signal system in (1) can be

interpolated by the following l local linear stochastic Poissonsignal systems

dx (t) = [Aix (t) +Biv1 (t)] dt+ Cix (t) dW (t)+Dix (t) dN (t)Y (t) = Qix (t) +Kiv2 (t) + Jix (t)w (t)+Mix (t)n (t)

s (t) = Gx (t) for i = 1, 2, ..., l(34)

To obtain a tractable mathematical interpretation of Y (t)in (34) [3], we introduce

y (t) =

∫ t

0

Y (s) ds (35)

Then we obtain the stochastic integral representation for(34)

dx (t) = [Aix (t) +Biv1 (t)] dt+ Cix (t) dW (t)+Dix (t) dN (t)dy (t) = [Qix (t) +Kiv2 (t)] dt+ Jix (t) dW (t)+Mix (t) dN (t)s (t) = Gx (t) for i = 1, 2, ..., l

(36)Based on the PLM method, the trajectory of the nonlinear

stochastic Poisson signal system in (1) could be approximatedby interpolating the trajectories of l local scheduled stochasticPoisson signal systems in (36) for some interpolation functionsµi (x) with i = 1, 2, ..., l [35] so that the nonlinear stochasticPoisson signal system in (1) can be approximated by thefollowing local interpolated scheduling systems:

dx (t) =l∑i=1

µi (x) Aix (t) dt+4f (x (t)) dt

+Biv1 (t) dt+4g (x (t))v1 (t) dt+ Cix (t) dW (t)+4h (x (t)) dW (t) + [Dix (t) +4i (x (t))] dN (t)

dy (t) =l∑i=1

µi (x) Qix (t) dt+4q (x (t)) dt

+Kiv2 (t) dt+4k (x (t))v2 (t) dt+ Jix (t) dW (t)+4j (x (t)) dW (t) + [Mix (t) +4m (x (t))] dN (t)

(37)where

f (x (t)) =

[l∑i=1

µi (x)Aix (t) +4f (x (t))

],

g (x (t)) =

[l∑i=1

µi (x)Bi +4g (x (t))

],

h (x (t)) =

[l∑i=1

µi (x)Cix (t) +4h (x (t))

],

i (x (t)) =

[l∑i=1

µi (x)Dix (t) +4i (x (t))

],

q (x (t)) =

[l∑i=1

µi (x)Qix (t) +4q (x (t))

],

k (x (t)) =

[l∑i=1

µi (x)Ki +4k (x (t))

],

j (x (t)) =

[l∑i=1

µi (x) Jix (t) +4j (x (t))

],

m (x (t)) =

[l∑i=1

µi (x)Mix (t) +4m (x (t))

],



6

and the interpolation functions µi (x) : x → [0 1] for i =

1, 2, ...l which have the property thatl∑i=1

µi (x) = 1 for all

x ∈ Φ.Based on the interpolation model in (37), the following

interpolated scheduling filter is proposed to represent the filterin (21) to deal with the H∞ robust filter design problem forthe nonlinear stochastic Poisson signal system in (1)

dx (t) =l∑

j=1

µj (x) (Ajx (t) dt+ Lj [dy (t)− dy (t)]) dt

dy (t) =l∑

j=1

µj (x)Qjx (t) dt

s (t) = Gx (t)(38)

where Lj is the local filter gain for the jth local filter.As it is not necessary to characterize a nonlinear stochastic

Poisson signal system by a high dimensional operating vectorx ∈ Φ, the scheduling space Z is introduced. If there existsa function Γ : RnΦ → RnZ that projects every vector in Φonto a lower dimensional scheduling space Z, then the systemdynamic behaviors can also be completely characterized bythe lower dimensional vector z ∈ Z (know as the schedulingvector). That is the interpolation functions µi (x) can bereplaced by scheduling function αi (z) for all i = 1, 2, ...,m.The relationship between operation regimes and schedulingregions are summarized in Table I. It is necessary to note thatΦi(operation regime) and Zi(scheduling region) are compactsets for all i where Φi =

Γ−1 (z) | z ∈ Zi

.

By introducing the scheduling space Z, the lower-dimensional vector z can be used to characterize thenonlinear stochastic Poisson signal system in (1) and theinterpolation function µ (x) in (37) and (38) can be replacedby the scheduling functions αi (z) : z → [0 1]. In thissituation, the scheduling filter in (38) could be rewritten as

dx (t) =l∑i=1

l∑j=1

l∑k=1

αi (z)αj (z)αk (z) (Aj − LjQk)

x (t) dt+ Lj [Qix (t) +4q (x (t))] dt+ LjKiv2 (t) dt+Lj4k (x (t))v2 (t) dt+ LjJix (t) dW (t)+Lj4j (x (t)) dW (t) + LjMix (t) dN (t)+Lj4m (x (t)) dN (t)

(39)Subsequently, the augmented system in (25) can be representedby the following filtering error dynamic system

dx (t) =l∑i=1

l∑j=1

l∑k=1

αi (z)αj (z)αk (z)Aijkx (t) dt

+4f (x (t)) dt+[Bij +4g (x (t))

]v (t) dt

+Cijx (t) dW (t) +4h (x (t)) dW (t) +Dijx (t) dN (t)+4ı (x (t)) dN (t)

e (t) = Gx (t)(40)

where

Aijk =

[Ai 0−LjQi Aj + LjQk

], Bij =

[Bi 00 −LjKi

]Cij =

[Ci 0−LjJi 0

], Dij =

[Di 0

−LjMi 0

]

4f (x (t)) =

4f (x (t))

−l∑

j=1

αj (z)Lj4q (x (t))

,4g (x (t)) =

4g (x (t))

−l∑

j=1

αj (z)Lj4k (x (t))

,4h (x (t)) =

4h (x (t))

−l∑

j=1

αj (z)Lj4j (x (t))

,and 4ı (x (t)) =

4i (x (t))

−l∑

j=1

αj (z)Lj4m (x (t))

.

Assumption 1: There exist some positive scaling bound-ing constants for the following approximation errors in thescheduling procedures

‖4f (x (t))‖2 ≤ e1 ‖x (t)‖2 ; ‖4h (x (t))‖2 ≤ e2 ‖x (t)‖2 ;

‖4i (x (t))‖2 ≤ e3 ‖x (t)‖2 ; ‖4q (x (t))‖2 ≤ e4 ‖x (t)‖2 ;

‖4j (x (t))‖2 ≤ e5 ‖x (t)‖2 ; ‖4m (x (t))‖2 ≤ e6 ‖x (t)‖2 ;[4g (x (t))4k (x (t))

] [4g (x (t))4k (x (t))

]T≤ εI

(41)

Remark 3. The higher the number l of scheduling functions,the lower the approximation error is. This leads to a trade-off between the approximation error and the computationcomplexity for the filter design. The number of schedulingfunctions should be chosen carefully so that the boundingconditions of the approximation errors in (41) do affect thesolvability of the H∞ robust filter design based on LMIsafterwards. Therefore, with adequate scheduling basis vectorsand scheduling functions, the approximation errors should notbe large enough to affect the solvability of filter gain.

Lemma 3. Let Mi be any matrix with appropriate dimensionand αi (z) be the scheduling interpolation function for the ithlocal system and P = PT > 0 . Then, we have(

l∑i=1

αi (z)Mi

)P

l∑j=1

αj (z)Mj

≤ l∑i=1

αi (z)MiPMi

(42)

With Lemma 3, the following theorem can be used toefficiently solve the robust filter design problem.

Theorem 2. For the augmented system in (40) under Assump-tion 1, if there exists a symmetric positive definite matrix so-lution P > 0 for the following two bilinear matrix inequalites(BMIs):

Πijk PBij CTijP DTijP PLj

? −γ2

2 I 0 0 0? ? −P2 0 0? ? ? − P

2λ 0

? ? ? ? −γ2

2ε I

< 0 (43)

andLTj PLj − βjI < 0 (44)



7

where

Πijk ,[ATijkP + PAijk

]+ λ

[DTijP + PDij

]+ (1 + λ)P

+GGT + Θ, Lj , diag (I6, Lj) , βj ∈ R+,

Θ , diag (β [(e1 + e4) + 2 (e2 + e5) + 3λ (e3 + e6)] I, 0) ,∀j = 1, ..., l.

then (i) the filtering error e (t) of system (40) in the casev (t) = 0 is asymptotical to zero in probability, and (ii) theH∞ filtering performance (26) or (27) is achieved with theprescribed disturbance filtering level γ > 0 in the case v (t) 6=0.

Proof. By HJI (28) in Theorem 1 and the augmented systemin (40), if there exists a positive Lyapunov function V (·) ∈C2 (Rnx) with V (0) = 0 such that the following HJI holds

Ψ (V (x (t))) = V Tx

[l∑i=1

l∑j=1

l∑k=1

αi (z)αj (z)αk (z)([Aijkx (t) + 4f (x (t))

])]+ 1

4γ2VTx

×

[l∑i=1

l∑j=1

αi (z)αj (z) Bij +4g (x (t))

]

×

[l∑

g=1

l∑h=1

αg (z)αh (z)(Bgh +4g (x (t))

)T]Vx

+xT (t) GT Gx (t) + 12

[l∑i=1

l∑j=1

αi (z)αj (z)

(Cijx (t) +4h (x (t))

)T ]Vxx

[l∑

g=1

l∑h=1

αg (z)

αh (z) Cgh +4h (x (t))]

+ λ

V

(x (t) +

l∑i=1

l∑j=1

αi (z)αj (z) Dijx (t) +4ı (x (t))

)−V (x (t)) < 0

(45), then the augmented system (40) is satisfied with the H∞robust filtering performance in (26) or (27).

Let the Lyapunov energy function be chosen as V (x (t)) =xT (t)Px (t). Substituting the Lyapunov energy function into(45), we can get

Ψ(xT (t)P x (t)

)=

l∑i=1

l∑j=1

l∑k=1

αi (z)αj (z)αk (z) xT (t)

×[2PAijk

]x (t) + 2xT (t)P4f (x (t)) + γ−2

×

l∑i=1

l∑j=1

l∑g=1

l∑h=1

αi (z)αj (z)αg (z)αh (z) xT (t)

×[PBijB

TghP

]x (t)

+ xT (t) GGT x (t)

+

l∑i=1

l∑j=1

l∑g=1

l∑h=1

αi (z)αj (z)αg (z)αh (z) xT (t)

×[CTijPCgh

]x (t)

+

l∑i=1

l∑j=1

αi (z)αj (z) xT (t)

×[24hT (x (t))PCij

]x (t)

+ γ−2

×

l∑i=1

l∑j=1

αi (z)αj (z) xT (t)P[2Bij4gT (x (t))

]x (t)

+γ−2

xT (t)P

[4g (x (t))4gT (x (t))

]P x (t)

+4hT (x (t))P4h (x (t))

+λ

l∑i=1

l∑j=1

l∑g=1

l∑h=1

αi (z)αj (z)αg (z)αh (z)

×xT (t)[DTijPDgh

]x (t)

+λ

l∑i=1

l∑j=1

αi (z)αj (z) 2xT (t)P[Dijx (t) +4ı (x (t))

]

+λ

l∑i=1

l∑j=1

αi (z)αj (z) 2xT (t) DTijP4ı (x (t))

+λ4ıT (x (t))P4ı (x (t))

By applying Lemma 3, we have

Ψ(xT (t)P x (t)

)≤

l∑i=1

l∑j=1

l∑k=1

αi (z)αj (z)αk (z)

xT (t)[2PAijk

]x (t) + 2xT (t)P4f (x (t))

+2γ−2xT (t)PBijBTijP x (t) + xT (t) GT Gx (t)

+2γ−2xT (t)P4g (x (t))4gT (x (t))P x (t)+2xT (t) CTijPCijx (t) + 24hT (x (t))P4h (x (t))+2λxT (t)PDijx (t) + λxT (t)P x (t)+2λxT (t) DT

ijPDijx (t) + 3λ4ıT (x (t))P4ı (x (t))(46)

≤l∑i=1

l∑j=1

l∑k=1

αi (z)αj (z)αk (z) xT (t)[ATijkP + PAijk

]+ λ

[DTijP + PDij

](1 + λ)P + GT G+ 2CTijPCij

+2λDTijPDij + 2γ−2PBijB

TijP

+2γ−2PLj

[4g (x (t))4k (x (t))

] [4g (x (t))4k (x (t))

]TLTj P

x (t)

+

[4f (x (t))4q (x (t))

]TLTj PLj

[4f (x (t))4q (x (t))

]+2

[4h (x (t))4j (x (t))

]TLTj PLj

[4h (x (t))4j (x (t))

]+3λ

[4i (x (t))4m (x (t))

]TLTj PLj

[4i (x (t))4m (x (t))

]< 0

(47)where Lj = diag (I6, Lj).

By the fact that LTj PLj ≤ βjI6 in (44) and schedulingerror bounds in (41), we get

Ψ(xT (t)P x (t)

)≤

l∑i=1

l∑j=1

l∑k=1

αi (z)αj (z)αk (z) xT (t)[ATijkP + PAijk

]+ λ

[DTijP + PDij

]+ (1 + λ)P + GT G+ 2CTijPCij + 2λDT

ijPDij

+2γ−2PBijBTijP + 2εγ−2PLjL

Tj P

x (t)

+xT (t)β [(e1 + e4) + 2 (e2 + e5) + 3λ (e3 + e6)]x (t) < 0(48)

The inequality in (48) holds if the following inequalitieshold[ATijkP + PAijk

]+ λ

[DTijP + PDij

]+ (1 + λ)P + GT G

+Θ + 2γ−2PBijBTijP + 2CTijPCij + 2λDT

ijPDij

+2εγ−2PLjLTj P < 0 for all i, j, k = 1, 2, ..., l

(49)Then, if the inequalities in (49) are satisfied, the H∞

filtering performance (26) or (27) will hold. By applying Schur



8

complement [3], the inequalities in (49) are equivalent to BMIsin (43).

Remark 4. In the scheduling filter design (38), we use BMIsin (43) and (44) to replace the HJI in (13) for simplicity inthe nonlinear state filter design in (38). The BMIs in (43) and(44) could be considered an equivalent solution to the HJI in(13) from the PLM scheduling viewpoint.

In general, it is still not easy to solve BMIs in (43) and (44)for robust scheduling filter design in (38).

Since the Lyapunov energy function based on the physicalmeaning should depend on both the real state of the nonlinearstochastic system (1) and the filtering error of the filteringerror system (40), for our design convenience, we can choosethe form of the Lyapunov energy function as V (x (t)) =xT (t)P x (t) where

P =

[P1 −P2

−P2 P2

]> 0 (50)

Then, BMIs (43) and (44) can be modified as equivalent LMIsbelow

Π1 ? ? ? ?Π2 Π3 ? ? ?

BTi P1 −BTi P2 −γ2I62 ? ?

KTi Y

Tj −KT

i YTj 0 −γ

2I62 ?

P1Ci + YjJi 0 0 0 −P1

2

−P2Ci − YjJi 0 0 0 P2

2P1Di + YjMi 0 0 0 0−P2Di − YjMi 0 0 0 0

P1 −P2 0 0 0Y Tj −Y Tj 0 0 0? ? ? ? ?? ? ? ? ?? ? ? ? ?? ? ? ? ?? ? ? ? ?−P2

2 ? ? ? ?0 −P1

2λ ? ? ?0 P2

2λ−P2

2λ ? ?

0 0 0 −γ2I62ε ?

0 0 0 0 −γ2I62ε

< 0

(51)and

−βjI6 ∗ ∗ ∗0 −βjI6 ∗ ∗P1 −Yj −P1 ∗−P2 Yj P2 −P2

< 0 (52)

respectively, where

Π11 ,(ATi + λDT

i

)P1 + P1 (Ai + λDi)

+(QTi + λMT

i

)Y Tj + Yj (Qi + λMi) +GTG

+ (1 + λ)P1 + β [(e1 + e4) + 2 (e2 + e5)] I6+β [3λ (e3 + e6)] I6,

Π21 , −ATj P2 −QTk Y Tj − P2 (Ai + λDi)−Yj (Qi + λMi)−GTG− (1 + λ)P2,

Π22 , ATj P2 + P2Aj +QTk YTj + YjQk

+GTG+ (1 + λ)P2,

and Yj , P2Lj .

Remark 5. Using the assertion of the Lyapunov energyfunction with P in (50), the BMIs in (43) and (44) can betransformed into the LMIs in (51) and (52), respectively, tomore easy solve for the H∞ robust scheduling filtering design.

Through the above analysis, with the choice of P in(50), the optimal H∞ scheduling filter design problem can bereframed as minimizing the filtering level γ for the followingeigenvalue problem [3]:

γ∗ 2 = minP1,P2

γ2

subject toP > 0, P1 > 0, P2 > 0, (51), and (52),

(53)

so that the optimal H∞ scheduling filter design problem in(53) could be solved by decreasing γ until no positive-definitesolution is achieved in the constrained optimization problem(53). Since the LMIs in (51) and (52) can be solved using theLMI toolbox in Matlab [26], the LMI-constrained optimizationproblem in (53) for the optimal H∞ robust filter design canalso be solved easily.

Based on the above analyses, we give the following designprocedure for the H∞ robust scheduling filtering design of anonlinear stochastic Poisson signal system, which is the mainresult of our paper.

Design Procedure:I. Give the linear combination matrix G in (21).II. Select the scheduling basis vector zi and construct l

linear models to approximate the nonlinear Poisson systemin (37) and the scheduling filter in (38).

III. Calculate the bounding constants for the modelingapproximation errors of the given nonlinear stochastic Poissonsignal system due to the scheduling procedures in (41), i.e. e1˜ e6 and ε in (41).

IV. Solve the LMI-constrained optimization problem in (53)to obtain γ∗, P > 0, P1 > 0, P2 > 0, βj > 0 and Lj = P−12 Yjfor the optimal H∞ robust filter design.

V. Establish the H∞ robust scheduling filter for the nonlin-ear stochastic Poisson signal system in (38).

IV. SIMULATION EXAMPLE

In this section, an incoming ballistic missile trajectoryrobust estimation problem is given to verify our results inSection III. Consider the incoming ballistic missile geometryshown in Fig. 1. The dynamic equations of motions for anincoming ballistic missile [45], [46] are obtained by classicalkinematic analysis and given as

x (t) =−ρg

√x (t)

2+ y (t)

2+ z (t)

2

2βx (t)

y (t) =−ρg

√x (t)

2+ y (t)

2+ z (t)

2

2βy (t)

z (t) =−ρg

√x (t)

2+ y (t)

2+ z (t)

2

2βz (t)− g

where (x (t) , y (t) , z (t)) are the radar-centered Cartesian co-ordinates of the incoming ballistic missile on the ground; g



9

is the gravity constant; β is the ballistic coefficient which isassumed to be a constant; ρ is the density of the atmosphereat the incoming ballistic missile position and can be approxi-mated by

ρ (z (t)) =

ρhe−αhz, ρh = 1.754, αh = 1.490× 10−4

if z (t) ≥ 9144 metersρle−αlz, ρl = 1.227, αl = 1.093× 10−4

if z (t) < 9144 meters

The overall nonlinear dynamic model of the incomingballistic missile system can be written in the following form

X (t) =[x (t) , y (t) , z (t) , −ρgξ(X)

2β x (t) , −ρgξ(X)2β y (t) ,

−ρgξ(X)2β z (t)− g

]T= f (X (t))

(54)where

X (t) = [x (t) , y (t) , z (t) , x (t) , y (t) , z (t)]T (55)

andξ (X) =

√x (t)

2+ y (t)

2+ z (t)

2.

To simplify the notations, in the following, we use

X = [x1, x2, x3, x4, x5, x6]T

to replace X (t) in Eq. (55). Suppose the incoming bal-listic missile suffers from both continuous random noiseh (X) dW (t) due to stochastic parameter fluctuation andPoisson jump discontinuous noise i (X) dN (t) due to thecheating(maneuvering) jets of the incoming ballistic missilethat generate swaggering side drifts to avoid being targeted byhostile missile with E dN (t) = 0.2dt, respectively. Further,a guidance law u (t) = KX = [−0.1x1 − x4, −0.1x2 − x5,−0.08x3 − x6, −0.01x4,−0.01x5, −0.01x6]T is employedto guide the missile to attack target.

Then, the nonlinear stochastic system for the incomingballistic missile with guidance can be written as

dX = [f (X) +KX + g (X)v1 (t)] dt+h (X) dW (t) + i (X) dN (t)

(56)

where

f (X) +KX =

−0.1x1−0.1x2−0.08x3−ρgξ(X)

2β x4 − 0.01x4−ρgξ(X)

2β x5 − 0.01x5( (2× 10−4

)x3 − ρgξ(X)

2β x6−0.01x6 − g

)

h (X) =

−0.005x1−0.005x2−0.004x3

−ρgξ(X)200β x4 − 0.0001x4−ρgξ(X)

200β x5 − 0.0001x5( (2× 10−6

)x3 − ρgξ(X)

200β x6−0.0001x6 − 0.01g

)

g (X) = I6, and i (X) =[0 0 0

x3350

x3350

0]T

Additionally, the measurement equation by the ground radarsystem, which is corrupted by external disturbance and con-tinuous and discontinuous noises, is given as

dy (t) = [q (X) + k (X)v2 (t)] dt+j (X) dW (t) +m (X) dN (t)

(57)

where q (X) = [x1 x2 x3 x4 x5 x6]T ; k (X) = I6; m (X) =[

0 0 0 x3

300x3

300 0]T

, and

j (X) =

−0.003x1−0.003x2−0.003x3

−ρgξ(X)200β x4 − 0.0001x4−ρgξ(X)

200β x5 − 0.0001x5( (2× 10−6

)x3 − ρgξ(X)

200β x6−0.0001x6 − 0.01g

)

.

It is necessary to note that in this simulation, the externaldisturbance v1 (t) and v2 (t) are assumed to be zero meanwhite noise with unit variance.

Suppose the optimal H∞ robust scheduling filter in (38)is employed by the radar system to estimate the incomingmissile trajectory X (t) from the measurement Y (t) in (55),i.e. s(t) = X. Using PLM scheduling interpolation method,we set the scheduling vector z =

[x3 x4 x5 x6

]T. The

scheduling basis vectors zi are given aszi =

[zi11 zi22 zi33 zi44

]T;

z11 = 13500, z21 = 20000, z31 = 25000, z41 = 35000;z51 = 45000, z61 = 58000, z12 = −1500, z22 = −1100;z13 = −1500, z23 = −1100, z14 = −1500, z24 = −1000;i1 = 1, .., 6; i2 = 1, 2; i3 = 1, 2; i4 = 1, 2 ;i = 8 (i1 − 1) + 4 (i2 − 1) + 2 (i3 − 1) + (i4 − 1) + 1so that we have 48 scheduling basis vectors. Then, the

scheduling functions are given as follows:

αi (z) =ρi (z)

48∑i=1

ρi (z)

(58)

where ρi (z) = ρi11 (z1) ρi22 (z2) ρi33 (z3) ρi44 (z4) ;

ρ11 (z1) =

1, if 0 ≤ z1 ≤ z111− (z1−z1

1)z21−z1

1, if z11 < z1 < z21

0, if z21 ≤ z1 ≤ 150000

;

ρ61 (z1) =

0, if 0 ≤ z1 ≤ z51(z1−z5

1)z61−z5

1, if z51 < z1 < z61

1, if z61 ≤ z1 ≤ 150000

;

ρk1 (z1) =

(z1−zk−1

1 )zk1−z

k−11

, if zk−11 < z1 ≤ zk1

1− (z1−zk1 )zk+11 −zk1

, if zk1 < z1 ≤ zk+11

0, elsefor k = 2, ..., 5

;

ρ12 (z2) =

1, if − 5000 ≤ z2 ≤ z121− (z2−z1

2)z22−z1

2, if z12 < z2 < z22

0, if z22 ≤ z2 ≤ 0

;



10

ρ22 (z2) =

0, if − 5000 ≤ z2 ≤ z12(z2−z1

2)z22−z1

2, if z12 < z2 < z22

1, if z22 ≤ z2 ≤ 0

;

ρ13 (z3) =

1, if − 5000 ≤ z3 ≤ z131− (z3−z1

3)z23−z1

3, if z13 < z3 < z23

0, if z23 ≤ z3 ≤ 0

;

ρ23 (z3) =

0, if − 5000 ≤ z3 ≤ z13(z3−z1

3)z23−z1

3, if z13 < z3 < z23

1, if z23 ≤ z3 ≤ 0

;

ρ14 (z4) =

1, if − 5000 ≤ z4 ≤ z141− (z4−z1

4)z24−z1

4, if z14 < z4 < z24

0, if z24 ≤ z4 ≤ 0

;

and ρ24 (z4) =

0, if − 5000 ≤ z4 ≤ z14(z4−z1

4)z24−z1

4, if z14 < z4 < z24

1, if z24 ≤ z4 ≤ 0The upper bounds of the PLM approximation errors in (41)

are also known to be

e1 = 0.082, e2 = 0.0082, e3 = 0.123, e4 = 0,e5 = 0.017, e6 = 0.123, and ε = 0.01

The initial condition in the simulation is assumed to be

XT (0) =[XT (0) XT (0)

]T=[XT (0) 0

]Twith

mX = EXT (0)

= [150000 210000 120000 − 2500 − 2500 − 2500]

, and

R =

[EX (0)XT (0)

?

EX (0)XT (0)

EX (0) XT (0)

]

=

[EX (0)XT (0)

0

0 0

]In this case solving the eigenvalue problem in (53) using the

LMI optimization toolbox in Matlab, we obtain γ∗ = 1.822

P1 = 0.85I6; P2 = 0.83I6

Fig. 2 to Fig. 7 show the trajectories of x1 (t) , x1 (t)to x7 (t) , x7 (t), respectively, which use the proposed H∞robust scheduling filter for estimation. The Poisson jumpprocess with jump intensity λ = 0.2 caused by maneuveringjets is shown in Fig. 8. Clearly, the proposed H∞ robustscheduling filter can efficiently decrease the state estimationerror caused by intrinsic continuous random noise, externaldisturbance and discontinuous cheating jet.

V. CONCLUSION

For the first time, since many nonlinear stochastic signalsystems may suffer from both continuous and discontinuousstochastic noises, the H∞ robust filtering design problemof nonlinear stochastic systems with both Poisson noise andWiener noise was investigated. For a class of nonlinearstochastic Poisson signal systems, the optimal H∞ robust

filter needs to solve a second-order nonlinear HJI-constrainedoptimization problem. Since there is still no efficient wayto solve second-order HJI-constrained optimization problemsdirectly, this study employed the PLM scheduling approachto simplify the H∞ robust filter design procedure so that theHJI was transformed into a set of LMIs, which were ableto be solved efficiently using the LMI toolbox in MATLAB.Finally, a robust trajectory estimation problem of an incomingballistic missile system under continuous parameter randomfluctuation and discontinuous maneuvering jets was given asan optimal H∞ robust filter design example for illustratingthe H∞ scheduling filter design procedure and to demonstratethe expected H∞ robust optimal filtering performance. Thus,the proposed H∞ robust scheduling filter design method isvery suitable for practical filtering applications in nonlinearstochastic Poisson signal systems with discontinuous randomnoise.

Table I. The relations between operation regimesand scheduling regions

Operation Space Scheduling SpaceΦ ⊂ ∪li=1Φi Z ⊂ ∪li=1Zi

operation regimes Φi operation region Zioperation basis

vector xi

scheduling basisvector zi = Γ (xi)

Relationsz = Γ (x) ⊂ ∪li=1Zi where x ∈ ∪li=1Φi

Φi =

Γ−1 (z) | z ∈ Zi

Fig. 1. The incoming ballistic missile geometry in simulation example[46](the relation between radar and incoming ballistic missile coordinates)where x axis denotes the downrange of the missile, y axis denotes the offrangeof the missile, and z axis denotes the altitude of the missile. The radar is setat the origin point of the Cartesian coordinate.

ACKNOWLEDGMENT

This work was supported by National Science Council undercontract MOST- 103-2745-E-007-001- ASP.



11

0 10 20 30 40 50 60 700

2

4

6

8

10

12

14

16

18x 10

4

Time (sec)

Dow

nran

ge (

m)

X

1

Estimation X1

Fig. 2. The downrange trajectory and its estimate by the proposed H∞scheduling filter.

0 10 20 30 40 50 60 700

0.5

1

1.5

2

2.5x 10

5

Time (sec)

Offr

ange

(m

)

X

2

Estimation X2

Fig. 3. The offrange trajectory and its estimate by the proposed H∞scheduling filter

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0

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13

[46] G. Chen, G. Chen, and S. H. Hsu, Linear stochastic control systems,CRC press, 1995.

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Bor-Sen Chen (F01, LF14) received the B.S. de-gree from Tatung Institute of Technology, Taipei,Taiwan, the M.S. degree from National CentralUniversity, Chungli,Taiwan, and the Ph.D. degreefrom the University of Southern California, LosAngeles, in 1970, 1973 and 1982, respectively. Hewas a Lecturer, Associate Professor, and Professor atTatung Institute of Technology from 1973 to 1987.He is currently the Tsing Hua University Professorof Electrical Engineering and Computer Science atNational Tsing Hua University, Hsinchu, Taiwan.

His current research interests are in control engineering, signal processingand systems biology. Dr. Chen has received the Distinguished Research Awardfrom the National Science Council of Taiwan four times. He is a ResearchFellow of the National Science Council of Taiwan and holds the excellentscholar Chair in engineering. He has also received the Automatic ControlMedal from the Automatic Control Society of Taiwan in 2001. He is anAssociate Editor of the IEEE Transactions on Fuzzy Systems from 2001 to2006 and Editor of the Asian Journal of Control. He is a Member of theEditorial Advisory Board of Fuzzy Sets and Systems and the InternationalJournal of Control, Automotion and Systems. He is the Editor in Chief ofInternational Journal of Fuzzy Systems from 2005 to 2008. He is a LifeFellow of IEEE.

Chien-Feng Wu received the B.S. degree in elec-trical engineering and the M.S. degree in electricalengineering from Chung Hua University, Hsinchu,Taiwan, R.O.C., in 2006 and 2008, respectively. Heis currently working toward the Ph.D. degree inelectrical engineering in the Laboratory of Controland Systems Biolog, Department of Electrical En-gineering, National Tsing Hua University, Hsin-chu,Taiwan. His current research interests include robustcontrol, fuzzy control, multi-objective optimizationand nonlinear stochastic systems.

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