Optimal H2 filtering for periodic linear stochastic systems with multiplicative white noise perturbations and sampled measurements

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Journal of the Franklin Institute 352 (2015) 5985–6010 www.elsevier.com/locate/jfranklin

Optimal H2 filtering for periodic linear stochastic systems with multiplicative white noise perturbations and sampled measurements Vasile Dragana, Samir Aberkaneb,n, Ioan-Lucian Popac Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O.Box 1-764, RO-014700 Bucharest, Romania Université de Lorraine, CRAN, UMR 7039, Campus Sciences, BP 70239, Vandoeuvre-les-Nancy Cedex 54506, France c Department of Exact Sciences and Engineering, “1 Decembrie 1918” University of Alba Iulia, Alba Iulia 510009, Romania

a

b

Received 15 May 2015; received in revised form 19 August 2015; accepted 19 October 2015 Available online 11 November 2015

Abstract This paper addresses the problem of optimal H2 filtering for a class of continuous-time periodic stochastic systems with periodic sampled measurements. The class of admissible filters consist of deterministic continuoustime periodic systems with finite jumps. The optimal solution of the considered optimization problem is obtained by integrating a suitable generalized continuous-time Riccati equation with finite jumps. To illustrate the proposed filtering strategy, we consider a problem of field monitoring where sensors are distributed on a rectangular region in order to estimate the state of a diffusion process. We consider that the sensors and the filter communicate over a communication channel and we assume that the communication channel induces some communication constraints. More specifically, we consider a medium access constraint under which the shared network can only accommodate a limited number of simultaneous communications between components. & 2015 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

n

Corresponding author. E-mail addresses: [email protected] (V. Dragan), [email protected] (S. Aberkane), [email protected] (I.-L. Popa). http://dx.doi.org/10.1016/j.jfranklin.2015.10.010 0016-0032/& 2015 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

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1. Introduction In this paper, we will address the problem of optimal filtering for a class of continuous-time periodic stochastic systems with discrete-time measurements (sampled measurements). More specifically, we consider a class of Itô differential equations with periodic coefficients and periodic sampled measurements. This class of stochastic models are very suitable for modeling systems subject to random parametric perturbations in applications that require data sampling (as it is the case of many actual processes). Some results concerning the filtering of linear stochastic systems (with continuous-time measurements) in the presence of multiplicative noise may be found for instance in [9,10,19,20]. H2 filtering results for linear time-invariant stochastic systems with sampled measurements were given in [7]. As reported in [18], the H2 sampled-filtering problem is closely related to the problem of H2 filtering under limited bandwidth constraints in networked control systems. Hence, sampled data filtering results may be extended to limited bandwidth systems. In [18], the authors adopted a deterministic point of view and considered the problem of H2 filtering of linear deterministic time-invariant systems under sampling. They first derived a discrete-time equivalent system that adequately incorporates the inter-sampling behavior of the original continuous-time plant. Then, they proposed a sub-optimal solution based on some sufficient linear matrix inequalities (LMIs) conditions. In our case, the class of admissible filters consist of deterministic continuous-time periodic systems with finite jumps and an arbitrary dimension of the state space (we will clarify later why we highlighted the deterministic nature of the proposed filter). In order to quantify the quality of the estimation achieved by an admissible filter, we introduce a performance criterion expressed in terms of the mean square of the deviation of the estimated signal zf(t) from the value of the signal z(t) which must be estimated. The filtering problem is then formulated in an H2 filtering setting. The proposed filtering procedure can be resumed as follows: (i) when no measurement is available, the state estimate is computed by integrating the deterministic part of the system model; (ii) when a measurement is received, the filter makes an impulsive correction on the state estimate. Such a procedure has been initially presented in [14] where the impulsive correction gain of the filter is obtained by integrating a continuous-time Riccati equation with finite jumps. Note however that in [14], the author does not consider the case with state-dependent noise affecting both the state-space equation and/or the measurement equation. Hence, a direct extension of the results in [14] to the case of stochastic systems with state-dependent noise will lead to a filter which is hardly implementable because it incorporates the multiplicative white noise perturbations which affect the given plant. That is why, for H2 type control problems and H2 type filtering problems for controlled systems with multiplicative white noise perturbations only suboptimal solutions were derived based on solvability of some systems of LMIs. For the readers convenience we refer to [10] and the references therein. Also, we can refer to [13] where the case of time-varying systems is considered. In our approach, the inconvenient due to the presence of the state multiplicative white noise perturbations in the state equation and/or in the measurement equation is overcome by involving the unique periodic solution of a suitable differential (algebraic) Lyapunov-type equation associated to the given plant. A similar technique was used in other filtering problems in [3,20]. In the present paper, an optimal solution is proposed and a corresponding deterministic (and

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hence practically implementable) filter is constructed. The optimal solution (the filter gain) of the considered optimization problem is here also obtained by integrating a suitable generalized continuous-time Riccati equation with finite jumps. The particularity of this Riccati equation is that its coefficients depend on the unique periodic solution of a suitably defined Lyapunov-type differential equation or an algebraic Lyapunov equation (in the case of time-invariant dynamic systems). This represents the key point in getting an optimal (and not only a sub-optimal) solution. It is worth mentioning that the simplest case where the methodology developed in this paper is applicable (while other approaches already existing in the literature cease to provide acceptable results) is the one where the signal which has to be estimated is generated by a deterministic linear time invariant system affected by additive white noise perturbations, while the available measurements are affected by state-multiplicative white noise perturbations and additionally the coefficients of the measured output are periodic sequences. This paper is organized as follows: Section 2 gives the mathematical model of the considered class of systems and describes the problem setting. Section 3 gives some preliminaries and auxiliary results. The main results are given in Section 4. Some numerical experiments are included in Section 5, while Section 6 contains some concluding
remarks. 

 Notations: In block matrices, ⋆ indicates symmetric terms: BAT CB ¼ BAT ⋆C ¼ ⋆A CB . The expression MN⋆ is equivalent to MNMT while M⋆ is equivalent to MMT. 2. Problem formulation Consider the time-varying linear system G having the state space representation given by 8 > < dxðtÞ ¼ A0 ðtÞxðtÞ dt þ A1 ðtÞxðtÞ dwðtÞ þ BðtÞ dvðtÞ; t Z 0 yðkhÞ ¼ ½C 0 ðkÞ þ C 1 ðkÞwd ðkÞxðkhÞ þ Dd ðkÞvd ðkÞ; ð1Þ > : zðtÞ ¼ C ðtÞxðtÞ z

where xðtÞ A Rn is the state of the system, yðkhÞA Rny are the measurements at the discrete-time instances t k ¼ kh, k A Zþ ¼ f0; 1; …;g (h40 being the sampling period); zðtÞ A Rnz is the signal to be estimated. In Eq. (1), fwðtÞgt Z 0 , fvðtÞgt Z 0 , fwd ðkÞgk A Zþ , fvd ðkÞgk A Zþ are stochastic processes on a given probability space ðΩ; F ; PÞ satisfying the following assumptions: (H1) (i) fwðtÞgt Z 0 is a 1-dimensional standard Wiener process with E½wðtÞ ¼ 0 and E½ðwðtÞ wðsÞÞ2  ¼ jt  sj, for all t; s A Rþ . (ii) fvðtÞgt Z 0 is a mv-dimensional standard Wiener process satisfying E½vðtÞ ¼ 0, E ½ðvðtÞ vðsÞÞðvðtÞ  vðsÞÞT  ¼ Vðt  sÞ, for all t Z s Z 0. (iii) fwd ðkÞgk Z 0 is a sequence of independent random variables with zero mean and variance 1. (iv) fvd ðkÞgk Z 0 is a sequence of mvd -dimensional independent random vectors with zero mean  and E vd ðkÞvTd ðkÞ ¼ V d , for all k A Zþ . (v) The processes fwðtÞgt Z 0 , fvðtÞgt Z 0 , fwd ðkÞgk Z 0 and fvd ðkÞgk Z 0 are independent stochastic processes. As usual, E½ stands for the mathematical expectation and the superscript T denotes the transposition of a matrix or of a vector. Regarding the coefficients of the system (1) we make the assumption:

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(H2) (i) t-Aj : R-Rnn , j¼ 0,1, t-B : R-Rnmv , t-C z : R-Rnz n are continuous and periodic matrix valued functions of period θ40. (ii) fCj ðkÞgk A Z  Rny n , j¼ 0,1, fDd ðkÞgk A Z  Rny mvd are periodic sequences of period θd A Z, θd 41. (iii) θθd h is a rational number. Remark 1. (a) Even if the system (1) works for t Z 0, the domain of definition of its coefficients can be extended in a natural way to the whole real axis due to the periodicity property. That is why we wrote t A R in the assumption (H2) (i) and k A Z in assumption (H2) (ii). (b) Since a constant function can be regarded as a periodic function of arbitrary period, we deduce that if in Eq. (1) we have Aj ðtÞ ¼ Aj , j¼ 0,1, BðtÞ ¼ B, C z ðtÞ ¼ Cz , t A R, then the assumption (H2) (iii) is automatically satisfied. In the general case, when the system (1) is timevarying, the assumption (H2) (iii) may be satisfied by a suitable choice of the sampling period h40. Our goal is to design a linear system Gf (usually named filter) that is fed to the input with the measured values y(kh), k ¼ 0; 1; …, generates a signal zf(t) which is the best estimation of the output z(t) of the given system (1). In this work, we consider Gf filters having the state space representation described by a linear system with finite jumps of the form 8 dx ðtÞ ¼ Af ðtÞxf ðtÞ dt; khot r ðk þ 1Þh > < f xf ðkhþ Þ ¼ Afd ðkÞxf ðkhÞ þ Bfd ðkÞyðkhÞ; k A Zþ ð2Þ > : z ðtÞ ¼ C ðtÞx ðtÞ; t Z 0: f f f The quality of the estimation achieved by a filter Gf is measured by the following performance criterion: Z
  1 t0 þτ  J Gf ¼ lim ð3Þ E jzðtÞ  zf ðtÞj2 dt: τ-1 τ t 0 Our investigation is performed under the assumption that the zero solution of the linear system dxðtÞ ¼ A0 ðtÞxðtÞ dt þ A1 ðtÞxðtÞ dwðtÞ is exponentially stable in mean square (ESMS). Regarding the filter Gf we make the assumption: (H3) (i) Af : R-Rnf nf , Cf : R-Rnz nf are continuous and periodic matrix valued functions of period θ.   (ii) Afd ðkÞ k A Z  Rnf nf , fBfd ðkÞgk A Z  Rnf nf are periodic sequences of period θd . (iii) The zero-solution of the linear system with finite jumps on Rnf 8 < dxf ðtÞ ¼ Af ðt Þxf ðt Þ; khot r ðk þ 1Þh dt ð4Þ : x ðkhþ Þ ¼ A ðkÞx ðkhÞ; k A Z f

fd

f

þ

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is exponentially stable, i.e. there exist βf Z 1, αf 40 such that     xf ðtÞ r βf e  αf ðt  t0 Þ xf ðt 0 Þ; for all t Z t 0 Z 0 and all xf ðt 0 ÞA Rnf . In the following F ad denotes the family of all linear systems with finite jumps of type (2) of arbitrary dimension nf Z 1 (of the corresponding state space) which satisfy the assumption (H3).  The problem of optimal filtering solved in this paper asks for the design of an admissible filter Gf which satisfies the following optimality condition:

 J Gf ¼ min JðGf Þ: ð5Þ Gf A F ad

In the time-invariant case, the optimal filtering problem (5) was solved in [7]. Coupling an admissible filter Gf A F ad to the system (1), we obtain the following system with finite jumps on the space Rn  Rnf : 8 dx ðtÞ ¼ A0cl ðtÞxcl ðtÞ dt þ A1cl ðtÞxcl ðtÞ dwðtÞ þ Bcl ðtÞ dvðtÞ; khot r ðk þ 1Þh > < cl xcl ðkhþ Þ ¼ ½Ad0cl ðkÞ þ Ad1cl ðkÞwd ðkÞxcl ðkhÞ þ Bdcl ðkÞvd ðkÞ; k A Zþ ð6Þ > : z ðtÞ ¼ zðtÞ z ðtÞ ¼ C ðtÞx ðtÞ; t Z 0 cl f cl cl  T where xcl ðtÞ ¼ xT ðtÞ xTf ðtÞ and !



A0 ðtÞ 0 A1 ðtÞ 0 BðtÞ A0cl ðtÞ ¼ ; Bcl ðtÞ ¼ ; Ccl ðtÞ ; A1cl ðtÞ ¼ 0 Af ðtÞ 0 0 0
 ¼ Cz ðtÞ  Cf ðtÞ ; Ad0cl ðkÞ ¼

In

0

Bfd ðkÞC 0 ðkÞ

Afd ðkÞ

! ;

Ad1cl ðkÞ ¼

0

0

Bfd ðkÞC1 ðkÞ 0

! ;

Bdcl ðkÞ

! 0 ¼ ð7Þ Bfd ðkÞDd ðkÞ  T For each t 0 Z 0 and x0cl ¼ xT0 xTf0 A Rn  Rnf , let xcl ðtÞ ¼ xcl ðt; t 0 ; x0cl Þ be the solution of the system (6) starting from x0cl at t ¼ t 0 . With this notation we may write successively        ð8Þ E jxðtÞ xf ðtÞj2 ¼ E Tr zcl ðtÞzTcl ðtÞ ¼ Tr C cl ðtÞY cl ðtÞC Tcl ðtÞ   where Tr½ represents the trace operator and Y cl ¼ E xcl ðtÞxTcl ðtÞ , t Z t 0 . So, Eq. (3) becomes Z
  1 t0 þτ  J Gf ¼ lim Tr Ccl ðtÞY cl ðtÞC Tcl ðtÞ dt: ð9Þ τ-1 τ t 0 In Section 4 we shall show that the right-hand side of Eq. (9) is well defined. Also, we shall express the value of the limit from Eq. (9) based on the unique θ-periodic solution of a suitable Lyapunov-type differential equation with finite jumps. We shall see that the value of this limit depends neither on the initial time t0 nor on the initial states x0cl .

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Proposition 1. Under the considered assumptions, Ycl(t), t Z t 0 defined via Eq. (8), is the unique solution of the problem with given initial values: 8 d > > Y cl ðt Þ ¼ A0cl ðt ÞY cl ðt Þ þ Y cl ðt ÞAT0cl ðt Þ þ A1cl ðt ÞY cl ðt Þ⋆ þ Bcl ðt ÞV⋆; khot r ðk þ 1Þh > > dt > > < 1 X þ Y ðkh Þ ¼ Adjcl ðkÞY cl ðkhÞ⋆ þ Bdcl ðkÞV d ⋆ cl > > > j¼0 > > > : Y ðt Þ ¼ x xT cl 0 0cl 0cl ð10Þ k ¼ k0 ; k0 þ 1; …; k0 being the integer defined by ðk0  1Þhot 0 r k0 h.

3. Preliminaries In this section we present several definitions and preliminary results which will be used to prove the main results of this work. For the readers convenient we shall point out several aspects specific to the time-varying case which cannot be derived mutatis-mutandis from the timeinvariant case. 3.1. Itô differential equations with finite jumps and associated Lyapunov differential equations Let

(

dxðtÞ ¼ M 0 ðtÞxðtÞ dt þ M 1 ðtÞxðtÞ dwðtÞ;

khot r ðk þ 1Þh

þ

xðkh Þ ¼ ðM d0 ðkÞ þ wd ðkÞM d1 ðkÞÞxðkhÞ

ð11Þ

where fwðtÞgt Z 0 , fwd ðkÞgk Z 0 are independent stochastic processes satisfying the assumption (H1) (i, iii). For each t Z 0, Ht stands for the s-algebra generated by the random variables w(s), wd(k), 0 r sr t, 0 r k such that kh r t. In Eq. (11), t-M j : R-Rmm , j ¼ 0,1, are bounded continuous matrix valued functions and fM dj ðkÞgk A Z  Rmm , j¼ 0,1, are bounded sequences. Let t 0 Z 0, x0 A Rm and k0 Z 0 be the integer defined by ðk0  1Þhot 0 r k0 h. Applying Theorem 5.2.1 in [17] on each interval ½t 0 ; k0 h and ½kh; ðk þ 1Þh, k Z k0 , we obtain that Eq. (11) has a unique solution xðtÞ ¼ xðt; t 0 ; x0 Þ with xðt 0 Þ ¼ x0 and with the following properties: (a) limt-khþ xðtÞ ¼ xkhþ ¼ ðM d0 ðkÞ þ wd ðkÞM d1 ðkÞÞxðkhÞ, for all k Z k0 ; (b) x(t) is Ht -measurable for all t Z t 0 ; (c) t-xðtÞ is left a.s. continuous in any t with t Z t 0 . Definition 1. We say that the zero solution of Eq. (11) is ESMS or equivalently, Eq. (11) is ESMS, if there exist β Z 1, α40 such that   E jxðt; t 0 ; x0 Þj2 r βe  αðt  t0 Þ jx0 j2 for all t Z t 0 Z 0, x0 A Rm .

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  Let YðtÞ ¼ E xðtÞxT ðtÞ , xðtÞ ¼ xðt; t 0 ; x0 Þ be a solution of Eq. (11). Applying Itô formula, we obtain that t↦YðtÞ is a solution of the linear differential equation with finite jumps 8 Y_ ðtÞ ¼ M 0 ðtÞYðtÞ þ YðtÞM T0 ðtÞ þ M 1 ðtÞYðtÞM T1 ðtÞ > > < 1 X ð12Þ þ Yðkh Þ ¼ M dj ðkÞYðkhÞM Tdj ðkÞ; k ¼ k0 ; k0 þ 1; … > > : j¼0

khot r ðk þ 1Þh, and satisfies the initial condition Yðt 0 Þ ¼ x0 xT0 , where k0 is the integer defined by ðk 0  1Þhot 0 r k0 h:

ð13Þ

One sees that Eq. (12) is a differential equation with finite jumps on the space S m of the m  m symmetric matrices. For any H A S m we denote Yðt; t 0 ; HÞ the solution of Eq. (12) satisfying the initial condition Yðt 0 ; t 0 ; HÞ ¼ H. Definition 2. Let I  R be a right unbounded interval. We say that the zero solution of the differential equation with jumps (12) is exponentially stable on the interval I, or equivalently the differential equation with jumps (12) is exponentially stable on the interval I if there exist β^ Z 1, ^ α40 (possible depending upon I) such that ^  t0 Þ ^  αðt J Yðt; t 0 ; HÞ J r βe JH J

for all t Z t 0 ; t; t 0 A I, H A S m , J  J being the spectral norm of a matrix.

Remark 2. It is obvious that if the linear differential equation with finite jumps (12) is exponentially stable on some right unbounded interval I then it is also exponentially stable on any subinterval I1  I, but it is not sure that it is exponentially stable on the whole real axis R. However, if t-M j ðtÞ; j ¼ 0; 1 are periodic functions of period θ and fM dj ðkÞgk A Z ; j ¼ 0; 1 are periodic sequences of period θd Z 1 and if θ; θd h satisfy the assumption (H2) (iii) then it can be proved that if the differential equation with finite jumps (12) is exponentially stable on a right unbounded interval I, then it is exponentially stable on the whole real axis.

Proposition 2. Under the assumption (H1) (i), (iii), the following are equivalent: (i) the Itô differential equation with finite jumps (11) is ESMS; (ii) the differential equation with finite jumps (12) is exponentially stable on the interval ½0; 1Þ. The proof may be done by direct calculations. The details are omitted for shortness. Let us consider the affine differential equation with finite jumps: 8 Y_ ðtÞ ¼ M 0 ðtÞYðtÞ þ YðtÞM T0 ðtÞ þ M 1 ðtÞYðtÞM T1 ðtÞ þ SðtÞ > > < 1 X þ Yðkh Þ ¼ M dj ðkÞYðkhÞM Tdj ðkÞ þ Sd ðkÞ; k AZ > > :

ð14Þ

j¼0

mm khot r ðk þ 1Þh, where , l ¼ 0; 1, S : R-S mare bounded and continuous matrix  M l : R-R   valued functions and M dj ðkÞ k A Z  Rmm , j ¼ 0,1, Sd ðkÞ k A Z are bounded sequences.

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Let Tðt; sÞ be the linear evolution operator on S m generated by the linear differential equation X_ ðtÞ ¼ LðtÞXðtÞ

ð15Þ

where LðtÞX ¼ M 0 ðtÞX þ XM T0 ðtÞ þ M 1 ðtÞXM T1 ðtÞ

ð16Þ

is the Lyapunov-type operator defined by the pair ðM 0 ðtÞ; M 1 ðtÞÞ. Invoking for example the developments from Chapter 3 of [4] we deduce that ðt; sÞ↦Tðt; sÞ : R  R-BðS m Þ verify the following initial value problem on BðS m Þ: 8 < d T ðt; sÞ ¼ Lðt ÞT ðt; sÞ dt ð17Þ : Tðs; sÞ ¼ I Sm

where I Sm is the identity operator on S m and BðS m Þ is the Banach algebra of linear operators on the space S m . The solutions of the differential equation (15) satisfy Xðt; t 0 ; X 0 Þ ¼ Tðt; t 0 ÞX 0 for all t; t 0 A R, X 0 A S m . Employing for example Corollary 3.8 from [5] we may conclude that the differential equation (15) generates a positive evolution on S m . This means that Tðt; sÞX Z 0, for all t Z s, if X Z 0. Based on the uniqueness of the solution of the problem with given initial value (17) we may infer that if t↦LðtÞ is a periodic function of period θ, then Tðt þ kθ; t 0 þ kθÞ ¼ Tðt; t 0 Þ for all t; t 0 A R; k A Z:

ð18Þ

Proposition 3. If the zero solution of linear differential equation with finite jumps of type (12) associated to Eq. (14) is exponentially stable on the whole real axis, then, the following holds: 

(i) the affine differential equation with jumps (14) has a unique solution Y ðÞ bounded on the whole real axis;  (ii) if SðtÞ Z 0, for all t A R and Sd ðkÞZ 0, k A Z, then Y ðtÞ Z  0 for all  t A R;   (iii) if t↦LðtÞ, t↦SðtÞ are periodic functions of period θ and M dj ðkÞ k A Z , j¼ 0,1, Sd ðkÞ k A Z  θ is a rational number, then t↦ Y ðtÞ is a are periodic sequences of period θd Z 1 and if θd h periodic function of period θ~ ¼ qθd h ¼ θqd where q, qd are coprime integer numbers such θ q ¼ . that θ d h qd

Proof. Consider the discrete-time equation on S m : Yðk þ 1Þ ¼ Ld ðkÞYðkÞ þ HðkÞ

ð19Þ

where Ld ðkÞ : S m -S m is defined by Ld ðkÞX ¼

1 X j¼0

 T ððk þ 1Þk; khÞ M dj ðkÞXM Tdj ðkÞ

ð20Þ

V. Dragan et al. / Journal of the Franklin Institute 352 (2015) 5985–6010

for all X A S m and

Z

5993

ðkþ1Þh

HðkÞ ¼ T ððk þ 1Þh; khÞSd ðkÞ þ

T ððk þ 1Þh; sÞSðsÞ ds:

ð21Þ

kh

One sees that for each k A Z, Ld ðkÞ is a linear and positive operator, that is Ld ðkÞX Z 0, if X Z 0. Let T d ðk; lÞ be the linear evolution operator defined by the discrete-time linear evolution Xðk þ 1Þ ¼ Ld ðkÞXðkÞ, that is T d ðk; lÞ ¼ Ld ðk  1ÞLd ðk  2Þ⋯Ld ðlÞ and T d ðk; lÞ ¼ I S m if k ¼ l:

if k4l; k; lA Z

By direct calculation one obtains that T d ðk; lÞY 0 ¼ Yðkh; lh; Y 0 Þ

ð22Þ

where t↦Yðt; lh; Y 0 Þ is the solution of linear differential equation of type (12) associated to Eq. (14) satisfying Yðlh; lh; Y 0 Þ ¼ Y 0 . The equality (22) allows us to deduce that J T d ðk; lÞJ r βe  αðk  lÞh JY 0 J ; for all k Z l, k; lA Z, Y 0 A S m , where β Z 1, α40 are  constants.  This shows that the sequence of linear operators Ld ðkÞ k A Z defines an exponentially stable evolution on S m . Employing Theorem 2.6 (i) from [6] we infer that the discrete-time affine equation (19) has a unique bounded on Z solution. This solution is given by 

X ðkÞ ¼

k1 X

T d ðk; l þ 1ÞHðlÞ:

l ¼ 1

From Eq. (21) we obtain that HðkÞZ 0 for all k A Z if SðtÞ Z 0, t A R and Sd ðkÞ Z 0, for all k A Z. Applying Theorem 2.6 (iv) from [6] we deduce that under this additional assumption we have  X ðkÞZ 0, for all k A Z. For each integer k we define 

Y ðkhþ Þ ¼

1 X j¼0



M dj ðkÞ X ðkÞM Tdj ðkÞ þ Sd ðkÞ:

If t A ðkh; ðk þ 1Þh we define 



Y ðtÞ ¼ Tðt; khÞ Y ðkhþ Þ þ

Z

ð23Þ

t

ð24Þ

Tðt; sÞSðsÞ ds: kh





Taking t ¼ ðk þ 1Þh in Eq. (24) we obtain, via Eqs. (19)–(21) and (23), that Y ðkhÞ ¼ X ðkÞ, for all k A Z. In this way, Eq. (23) becomes 

Y ðkhþ Þ ¼

1 X j¼0



M dj ðkÞ Y ðkhÞM Tdj ðkÞ þ Sd ðkÞ;

k A Z:

Differentiating Eq. (24) we obtain  d  Y ðt Þ ¼ Lðt Þ Y ðt Þ þ Sðt Þ; khot r ðk þ 1Þh dt  which shows that Y ðtÞ, t A R is a globalsolution of Eq. (14). FromEqs. (23) and (24) together with the boundedness of the sequence f X ðkÞgk A Z we obtain that J Y ðtÞ J r c, for all t AR. This  shows that Y ðtÞ is a global and bounded solution of Eq. (14). If Y 1 ðtÞ, t A R, would be another global and bounded solution of Eq. (14) then, from the uniqueness of the bounded solution of Eq.

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(19) we deduce that Y 1 ðkhÞ ¼ Y ðkhÞ, k A Z, which leads to Y 1 ðtÞ ¼ Y ðtÞ, khot rðk þ 1Þh, k A Z, which confirms the uniqueness of the bounded on R solution of Eq. (14). Thus (i) is proved.  (ii) follows immediately from Eqs. (23) and (24) together with X ðkÞZ 0. In order to prove (iii) let us assume that M l ðÞ, l¼ 0,1, SðÞ are continuous and θ-periodic functions and fM dj ðkÞgk A Z , j ¼ 0,1, fSd ðkÞgk A Z are periodic sequences of period θd . Assume also θ is a rational number. Let q; qd be coprime natural numbers such that θθd h ¼ qq . Let that d θd h ~θ ¼ qd θ ¼ qqd h and θ^ ¼ qθd . Using Eqs. (18) and (20) one obtains that Ld ðk þ θÞ ^ ¼ Ld ðkÞ,   ^ ¼ HðkÞ, for all k A Z. Invoking Theorem 2.6 (ii) in [6] we infer that X ðk þ θÞ ^ ¼ X ðkÞ Hðk þ θÞ for all k A Z. This allows us to deduce via Eq. (23) that   ^ þ ¼ Y ðkhþ Þ; k A Z: Y ðk þ θÞh ð25Þ 



Also, the equality Y ðkhÞ ¼ X ðkÞ, k A Z, yields   ^ Y ðk þ θÞh ¼ Y ðkhÞ for all k A Z: Let t A R be arbitrary such that t a jh, jA Z. This means that there exists k A Z with the property ^ ~ khotoðk þ 1Þh. Therefore, ðk þ θÞhot þ θoðk þ θ^ þ 1Þh. Writing Eq. (24) for k replaced by ^ ~ k þ θ and t replaced by t þ θ one obtains via Eq. (25) combined with a simple changing of   ~ ¼ Y ðtÞ, t A R. Thus, the proof is completed.□ variables of integration, that Y ðt þ θÞ

Remark 3. The assumption that all coefficients of the differential equation with finite jumps (14) are defined on the whole real axis is essential in order to have a unique bounded solution. Let us remark that if the coefficients of Eq. (14) would be defined on a left bounded interval ½a; 1Þ, then all its solutions are bounded if the zero solution of the corresponding linear differential equation of type (12) is exponentially stable. Let us assume that M l ðtÞ ¼ M l and SðtÞ ¼ S, for all t A R. Under these conditions we may assume without loss of generality that M l ðÞ and SðÞ are periodic functions of period θ ¼ θd h. Proposition 3 (iii) allows us to conclude that if M l ðtÞ ¼ M l , l¼ 0,1, and SðtÞ ¼ S for all t AR, and if fM dj ðkÞgk A Z , j¼ 0,1, fSd ðkÞgk A Z are periodic sequences of period θd , then the unique bounded on R solution of the differential equation (17) is periodic with period θd h. Furthermore, if θd ¼ 1 that is M dj ðkÞ ¼ M dj , j¼ 0,1, Sd ðkÞ ¼ Sd , for all k A Z, then the unique bounded on R solution of differential equation (17) is a periodic function of period h, and thus we recover the result proved in [7]. At the end of this subsection, let us consider the Itô differential equation (11) in the special case M d1 ðkÞ ¼ 0, M d0 ðkÞ ¼ I m for all k A Z. In this case, the solutions xðtÞ ¼ xðt; t 0 ; x0 Þ of Eq. (11) will also solve the Itô differential equation dxðtÞ ¼ M 0 ðtÞxðtÞ dt þ M 1 ðtÞxðtÞ dwðtÞ;

t Z 0:

ð26Þ

The linear differential equation with finite jumps (12) reduces to Eq. (15). The result stated in Proposition 2 becomes:

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Corollary 4. Under the considered assumptions, the following are equivalent: (i) the zero solution of the differential equation (26) is ESMS; (ii) the zero solution of the differential equation (15) is exponentially stable on ½0; 1Þ. Furthermore, if t-M l ðtÞ; l ¼ 0; 1, are periodic functions of period θ40 then the assertions (i) and (ii) from above are equivalent to: (iii) If the zero solution of the linear differential equation (15) is exponentially stable on the whole real axis.

Let us consider now the affine differential equation Y_ ðtÞ ¼ LðtÞYðtÞ þ SðtÞ

ð27Þ

where L : R-BðS m Þ is the operator valued function defined via Eq. (16) and S : R-S m is a bounded and continuous matrix valued function. Applying Theorem 2.3.7 in [8] we have: Proposition 5. If the zero solution of the linear differential equation (15) is exponentially stable on the whole real axis, then the affine differential equation (27) has a unique bounded on R solution. This solution is given by Z t  Y ðtÞ ¼ Tðt; sÞSðsÞ ds: 1 

If LðÞ, SðÞ are periodic functions of period θ then, Y ðÞ is also a periodic function of the same period θ. Moreover, if SðtÞ Z 0, t A R, then Y ðtÞZ 0, t A R.

3.2. A class of differential equations with finite jumps of Riccati type An important role in the derivation of the state space representation of the optimal filter is played by a periodic solution of the following differential equation with finite jumps: (_ Y ðtÞ ¼ A0 ðtÞYðtÞ þ YðtÞAT0 ðtÞ þ QðtÞ
1 Yðkhþ Þ ¼ YðkhÞ YðkhÞC T0 ðkÞ  Rd ðkÞ þ C0 ðkÞYðkhÞCT0 ðkÞ ⋆ þ Qd ðkÞ; k A Z ð28Þ khot r ðk þ 1Þh, where the function t↦A0 ðtÞ and the sequence fC d ðkÞgk A Z occur in the state space representation of the system (1) and consequently they satisfy the assumption (H2). In Eq. (28), Q : R-S n is a continuous and θ-periodic function and fQd ðkÞgk A Z A S n , fRd ðkÞgk A Z A S ny are periodic sequences of period θd . We also assume that Rd ðkÞZ 0, Qd ðkÞZ 0 for all k A Z. A function Y : R-S n left continuous in each t 0 A R is a global solution of the differential equation (28) if Rd ðkÞ þ C 0 ðkÞYðkhÞC T0 ðkÞ is invertible for all k A Z and satisfies Eq. (28) on each interval ðkh; ðk þ 1ÞhÞ.

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A global solution Y s ðÞ is named the stabilizing solution of the differential equation with finite jumps (28) if the zero solution of the linear differential equation with finite jumps on Rn ( x_ ðtÞ ¼ A0 ðtÞxðtÞ; khot r ðk þ 1Þh ð29Þ xðkhþ Þ ¼ ðI n þ K s ðkÞC 0 ðkÞÞxðkhÞ; k A Z is exponentially stable, where K s ðkÞ ¼  Y s ðkhÞC T0 ðkÞðRd ðkÞ þ C 0 ðkÞY s ðkhÞC T0 ðkÞÞ  1 : In the developments of this paper, we need the stabilizing solution of the differential equation with finite jumps (28) satisfying the sign condition Rd ðkÞ þ C 0 ðkÞY s ðkhÞC T0 ðkÞ40 for all k A Z

ð30Þ

In this subsection we provide a set of conditions which guarantee the existence of the stabilizing solution of Eq. (28) which satisfy the condition (30). If YðÞ is a global solution of the differential equation with jumps (28) then, for any k A Z and any khot r ðk þ 1Þh we have Z t YðtÞ ¼ ϕ0 ðt; khÞYðkhþ ÞϕT0 ðt; khÞ þ ϕ0 ðt; sÞQðsÞϕT0 ðt; sÞ ds ð31Þ kh

where ϕ0 ðt; sÞ is the fundamental matrix solution of the differential equation on Rn x_ ðtÞ ¼ A0 ðtÞxðtÞ;

t A R:

ð32Þ

Taking t ¼ ðk þ 1Þh in Eq. (31) and using the second equation of Eq. (28) we deduce that the sequence fYðkhÞgk A Z is a global solution of the discrete-time forward Riccati equation (DTRE)
1 Yðk þ 1Þ ¼ AðkÞYðkÞ⋆ AðkÞYðkÞC T0 ðkÞ  Rd ðkÞ þ C0 ðkÞYðkÞCT0 ðkÞ ⋆ þ Sd ðkÞ ð33Þ where we denoted ( AðkÞ ¼ ϕ0 ððk þ 1Þh; khÞ Sd ðkÞ ¼ ϕ0 ððk þ 1Þh; khÞQd ðkÞ⋆ þ

R ðkþ1Þh kh

ϕ0 ððk þ 1Þh; sÞQðsÞ⋆ ds

If θθd h ¼ qq , q, qd being two coprime natural numbers, we define, as before, θ^ ¼ qθd , d ^ θ~  ¼ qd θ ¼ 0 ðt þ lθ; s þ lθÞ ¼ ϕ0 ðt; sÞ, for all t; sA R and all lA Z we deduce that θh. Since ϕ ^ ^ A k þ θ ¼ AðkÞ, Sd k þ θ ¼ Sd ðkÞ, for all k A Z. Taking into account that we also have,   Rd k þ θ^ ¼ Rd ðkÞ and C0 k þ θ^ ¼ C 0 ðkÞ we infer that the DTRE (33) has periodic ^ global solution fYs ðkÞgk A Z of DTRE (33) is a stabilizing solution if coefficients of period θ.A the zero solution of the discrete-time closed-loop system xðk þ 1Þ ¼ ðAðkÞ þ Ks ðkÞC 0 ðkÞÞxðkÞ is exponentially stable, where Ks ðkÞ ¼  AðkÞYs ðkÞCT0 ðkÞðRd ðkÞ þ C 0 ðkÞYs ðkÞCT0 ðkÞÞ  1 Based on the uniqueness of the bounded and stabilizing solution of DTRE (33) we obtain: Proposition 6. Under the considered assumptions the following are true:

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~ (i) if Ys(t), t A Z is the unique, stabilizing and θ-periodic solution of the differential equation  with jumps (28), then the sequence f Y ðkÞgk A Z defined by Y ðkÞ ¼ Y s ðkhÞ is the stabilizing ^ and θ-periodic solution of DTRE (33); ^ (ii) if fYs ðkÞgk A Z is the stabilizing and θ-periodic solution of DTRE (33) we define
1 Y s ðkhþ Þ ¼ Ys ðkÞ Ys ðkÞCT0 ðkÞ Rd ðkÞ þ C0 ðkÞYs ðkÞC T0 ðkÞ ⋆ þ Qd ðkÞ and þ

Z

Y s ðtÞ ¼ ϕ0 ðt; khÞY s ðkh Þ⋆ þ

t

ϕ0 ðt; sÞQðsÞ⋆ ds;

khot r ðk þ 1Þh; k A Z:

kh

~ The matrix valued function defined in this way is the stabilizing and θ-periodic solution of Eq. (28). The proof may be done by direct calculations. The details are omitted for shortness. The next result provides a set of conditions which guarantee the existence of the stabilizing ^ and θ-periodic solution of DTRE (33). Proposition 7. Assume that: ^ (a) the coefficients of DTRE (33) are periodic sequences of period θ; (b) the zero solution of the differential equation (32) is exponentially stable. Under these conditions the following are equivalent: ^ (i) the DTRE (33) has a unique stabilizing and θ-periodic solution fYs ðkÞgk A Z which satisfies the sign condition Rd ðkÞ þ C0 ðkÞYs ðkÞCT0 ðkÞ40;

8 k AZ:

ð34Þ

(ii) there exist the positive definite matrices Y^ ðjÞA S n , 0 r jr θ^  1 satisfying the following system of LMIs: ! ΘðjÞ AðjÞYðjÞCT0 ðjÞ ^ ¼ Y^ ð0Þ; 0r jr θ^  1: 40; Y^ ðθÞ ⋆ RðjÞ þ C 0 ðjÞY^ ðjÞCT0 ðjÞ where ΘðjÞ ¼ AðjÞY^ ðjÞ⋆ þ Sd ðjÞ  Y^ ðj þ 1Þ The proof follows immediately applying Theorem 5.6 from [6] in the special case of the dual equation T

T





T





xðkÞ ¼ A ðkÞxðk þ 1Þ⋆ A ðkÞxðk þ 1Þ B ðkÞð R d ðkÞ þ B ðkÞxðk þ 1Þ B ðkÞÞ  1 ⋆ þ MðkÞ 

T







where A ðkÞ ¼ A ð kÞ, B ðkÞ ¼ C T0 ð kÞ, R d ðkÞ ¼ Rd ð kÞ, MðkÞ ¼ Sd ð kÞ, k A Z. The details are omitted. Here we mention that the exponential stability of thesystem (32) guarantees the exponential stability ofthe discrete-time linear equation xðk þ 1Þ ¼ A ðkÞxðkÞ. This fact allows us  to deduce that ð A ðÞ; B ðÞÞ is stabilizable. From Proposition 6 one sees that for numerical computation of the unique stabilizing and ~ θ-periodic solution of the differential equation with finite jumps (28) it is sufficient to compute

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^ ^ the stabilizing and θ-periodic solution of DTRE (33). To compute the stabilizing and θperiodic solution of the discrete-time Riccati equation of type (33) satisfying the sign condition (34) there are numerous numerical methods either iterative methods or methods based on invariant subspaces of some suitable matrix pencils. For details see e.g. [21] or Chapter 12 from [1].

4. The main result 4.1. The value of the performance measure The development in this section is done under the following assumption: (H4) The zero solution of the Itô differential equation dxðtÞ ¼ A0 ðtÞxðtÞ dt þ A1 ðtÞxðtÞ dwðtÞ

ð35Þ

(obtained from Eq. (1) for BðtÞ  0) is ESMS. Consider the linear system with finite jumps ( dxcl ðtÞ ¼ A0cl ðtÞxcl ðtÞ dt þ A1cl ðtÞxcl ðtÞ dwðtÞ

ð36Þ

xcl ðkhþ Þ ¼ ðAd0cl ðkÞ þ wd ðkÞAd1cl ðkÞÞxcl ðkhÞ khot r ðk þ 1Þh, k A Z, obtained from Eq. (6) taking BðtÞ  0, Dd ðkÞ  0.

Proposition 8. Under the assumptions (H1)–(H4) for each admissible filter Gf the corresponding system (36) is ESMS.  T Proof. Let Gf be an admissible arbitrary filter and t 0 A R, x0cl ¼ xT0 xTf0 A Rn  Rnf . Let xcl ðtÞ ¼ xcl ðt; t 0 ; x0cl Þ be the corresponding solution of Eq. (6) with the initial condition xcl ðt 0 Þ ¼ x0cl . According to Definition 1, we have to show that there exist βf Z 1, αf 40 not depending upon t0 and x0cl , such that   E jxcl ðtÞj2 r βf e  αf ðt  t0 Þ jx0cl j2 ; 8 t Z t 0 Setting xcl ðtÞ ¼ ðxT ðtÞ xTf ðtÞÞT we obtain the following partition of Eq. (6): 8 dxðtÞ ¼ A0 ðtÞxðtÞ dt þ A1 ðtÞxðtÞ dwðtÞ > > > > < dxf ðtÞ ¼ Af ðtÞxf ðtÞ dt; khot r ðk þ 1Þh xðkhþ Þ ¼ xðkhÞ > > > > : xf ðkhþ Þ ¼ Afd ðkÞxf ðkhÞ þ Bfd ðkÞðC0 ðkÞ þ wd ðkÞC 1 ðkÞÞxðkhÞ

ð37Þ

k ¼ k0 ; k0 þ 1; …; k0 being defined as in Eq. (13). From the first and the third equation of Eq. (37) we deduce that x(t), t Z t 0 , is the solution of Itô differential equation (35). Based on the assumption (H4) we infer that there exist β1 Z 1, α1 40 such that     ð38Þ E jxðtÞj2 r β1 e  α1 ðt  k0 hÞ E jxðk 0 hÞj2 ; 8 t Z k 0 h Further, from the other two equations in Eq. (37) we obtain xf ððk þ 1ÞhÞ ¼ M 0 ðkÞxf ðkhÞ þ ½M 1 ðkÞwd ðkÞM 2 ðkÞxðkhÞ;

k Z k0

ð39Þ

where M 0 ðkÞ ¼ ϕf ððk þ 1Þh; khÞAfd ðkÞ and, for j ¼ 1, 2, M j ðkÞ ¼ ϕf ððk þ 1Þh; khÞBfd ðkÞC j ðkÞ, and,

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for j¼ 1, 2, ϕf ðt; sÞ being the fundamental matrix solution of the linear differential equation on Rnf x_ f ðtÞ ¼ Af ðtÞxf ðtÞ: From the exponential stability of the corresponding system (4) one obtains that there exist β2 Z 1, α2 40 such that the solutions of the discrete-time linear system ξðk þ 1Þ ¼ M 0 ðkÞξðkÞ satisfy jξðkÞj rβ2 e  α2 ðk  k0 Þ jξðk0 Þj. Reasoning as in the proof of Theorem 3.18 from [6] we obtain that the solution of Eq. (39) satisfies ! k1     X   2  α3 ðk  k0 Þh 2  α3 ðk  j  1Þh 2 E jxf ðkhÞj r β3 e E jxf ðk 0 hÞj þ e E jxðjhÞj ð40Þ j ¼ k0

Without lost of generality we may assume that α3 A ð0; α1 Þ, α1 being involved in Eq. (38). Plugging Eq. (38) in Eq. (40) we obtain after some computations that  
    E jxf ðkhÞj2 r β4 e  α3 ðk  k0 Þh E xf ðk 0 hÞj2 þ E jxðk 0 hÞj2 ; ð41Þ for all k Z k 0 . Since   E jxðk 0 hÞj2 r eγh jx0 j2 and

 (  E jxf ðk0 hÞj2 r eγ f h jxf 0 j2 ;     E jxf ðtÞj2 r eγf h μf E jxf ðkhÞj2 ;

where γ40, γ f 40, μf 40 are constants, we deduce from Eq. (41) that  
 E jxf ðtÞj2 r β5 e  α3 ðt  t0 Þ jx0 j2 þ jxf j2 and    E jxðtÞj2 r β1 e  α1 ðt  t0 Þ jx0 j2 ; for all t Z t 0 , so the proof is completed.□ Taking BðtÞ ¼ 0, t A R, Dd ðkÞ ¼ 0, k A Z in Eq. (10) we obtain the following linear differential equation with finite jumps on S nþnf 8 Y_ cl ðtÞ ¼ A0cl ðtÞY cl ðtÞ þ Y cl AT0cl ðtÞ þ A1cl ðtÞY cl ðtÞ⋆ > > < 1 X ð42Þ þ Y ðkh Þ ¼ Adjcl ðkÞY cl ðkhÞATdjcl ðkÞ; k A Z > cl > : j¼0 khot r ðk þ 1Þh. Specializing the results proved in Proposition 2 and Proposition 3 to the differential equation (10) and (42) we obtain: Corollary 9. Under the assumption (H1)–(H4), for each admissible filter Gf the following hold:

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(i) the solution of the linear differential equation with finite jumps (42) satisfy JY cl ðtÞJ r βe  αðt  t0 Þ JY cl ðt 0 ÞJ

ð43Þ

for all t Z t 0 , t; t 0 A R, where β Z 1, α40 are constants; (ii) the affine differential equation with finite jumps (10) has a unique bounded on R solution   ^ and satisfies Y cl ðtÞ Z 0, for all Y cl ðÞ. This solution is a periodic function of period θ~ ¼ θh t A R. The next result provides the value of the performance measure (3) computed based on the ~ unique θ-periodic solution of the differential equation with finite jumps (10). Theorem 10. Under the assumptions (H1)–(H4), the value of the performance (3) achieved by an admissible filter Gf is given by Z i 
 1 θ~ h J Gf ¼ Tr Ccl ðtÞ Y cl ðtÞC Tcl ðtÞ dt: ð44Þ θ~ 0

Proof. According to Corollary 9 (ii) we infer that under the assumptions (H1)–(H4) the affine ~ differential equation with jumps (10) has a unique θ-periodic positive semi-definite solution  Y ðÞ. This allows us to rewrite Eq. (9) as follows: Z t0 þτ h i 
 Tr C cl ðtÞ Y cl ðtÞC Tcl ðtÞ dt J Gf ¼ lim τ-1

t0

1 þ lim τ-1 τ 

Z

t 0 þτ

h  i  Tr C cl ðtÞ Y cl ðtÞ Y cl ðtÞ ⋆ dt

ð45Þ

t0

Since t↦Y cl ðtÞ Y cl ðtÞ is a solution of Eq. (42) we obtain via Eq. (43) that 



JY cl ðtÞ Y cl ðtÞJ r βe  αðt  t0 Þ Jx0cl xT0cl  Y cl ðt 0 ÞJ : Hence 1 τ-1 τ

Z

lim

t0

t 0 þτ

h  i  C cl ðtÞ Y cl ðtÞ  Y cl ðtÞ C Tcl ðtÞ dt ¼ 0

Plugging the last equality in Eq. (45) we get Z i 
 1 t0 þτ h J Gf ¼ lim Tr C cl ðtÞ Y cl ðtÞCTcl ðtÞ dt τ-1 τ t 0 h i  Since y↦ C cl ðtÞ Y cl ðtÞC Tcl ðtÞ is a bounded function we may infer that Z Z t0 h i i   1 t0 þτ h lim Tr Ccl ðtÞ Y cl ðtÞ⋆ dt ¼ lim Tr C cl ðtÞ Y cl ðtÞ⋆ dt ¼ 0 τ-1 0 τ-1 τ τ for all t 0 A R. Thus Eq. (46) becomes Z i 
 1 τ h Tr C cl ðtÞ Y cl ðtÞC Tcl ðtÞ dt J Gf ¼ lim τ-1 τ 0

ð46Þ

V. Dragan et al. / Journal of the Franklin Institute 352 (2015) 5985–6010

Further,

Z

h i  Tr C cl ðt Þ Y cl ðt Þ⋆ dt ¼ 0 h i  because t↦Tr C cl ðtÞ Y cl ðtÞC Tcl ðtÞ is a completed.□ 1 lim τ-1 τ

τ

1 θ~

Z

θ~

6001

h i  Tr Ccl ðt Þ Y cl ðt Þ⋆ dt

0

~ Thus the proof is periodic function with period θ.

4.2. The state space representation of the optimal filter 

~ Let Gf be an arbitrary admissible filter and Y ðÞ be the unique θ-periodic solution of the corresponding affine differential equation (10). Let ! Y 11 ðtÞ Y 12 ðtÞ Y T12 ðtÞ Y 22 ðtÞ 

be the partition of the matrix Y cl ðtÞ such that Y 11 ðtÞ A S n and Y 22 A S nf , t A R. Eq. (10) satisfied  by Y cl ðÞ has the partition 8 Y_ 11 ðtÞ ¼ A0 ðtÞY 11 ðtÞ þ Y 11 ðtÞAT0 ðtÞ þ A1 ðtÞY 11 ðtÞ⋆ þ BðtÞV⋆ > > > > > > Y_ 12 ðtÞ ¼ A0 ðtÞY 12 ðtÞ þ Y 12 ðtÞATf ðtÞ > > > > > Y_ 22 ðtÞ ¼ Af ðtÞY 22 ðtÞ þ Y 22 ðtÞAT ðtÞ; khot r ðk þ 1Þh > f > > > > < Y 11 ðkhþ Þ ¼ Y 11 ðkhÞ ð47Þ Y 12 ðkhþ Þ ¼ Y 11 ðkhÞC T0 ðkÞBTfd ðkÞ þ Y 12 ðkhÞATfd ðkÞ > > > > > > Y ðkhþ Þ ¼ Bfd ðkÞC0 ðkÞY 11 ðkhÞC T0 ðkÞBTfd ðkÞ þ Afd ðkÞY T12 ðkhÞCT0 ðkÞBTfd ðkÞ > > 22 > > > þ Bfd ðkÞC0 ðkÞY 12 ðkhÞATfd ðkÞ þ Afd ðkÞY 22 ðkhÞATfd ðkÞ > > > > > : þ Bfd ðkÞC1 ðkÞY 11 ðkhÞCT1 ðkÞBTfd ðkÞ þ Bfd ðkÞDd ðkÞV d DTd ðkÞBTfd ðkÞ Employing the first and the fourth equation from Eq. (47) we may conclude that t↦Y 11 ðtÞ is a C1 function and it is a periodic solution of period θ~ of the differential equation Y_ ðtÞ ¼ A0 ðtÞYðtÞ þ YðtÞAT0 ðtÞ þ A1 ðtÞYðtÞ⋆ þ BðtÞV⋆ ð48Þ This differential equation is of type (27). Invoking Proposition 5 we may conclude that under the assumptions (H1), (H2) and (H4) the affine differential equation (48) has a unique bounded on R ~ solution, and additionally, this solution is periodic with period θ. ~ In the sequel, the unique θ-periodic solution of Eq. (48) will be denoted by Pc ðÞ. Let us introduce the differential equation with finite jumps of Riccati type 8  < Y_ ðtÞ ¼ A0 ðtÞYðtÞ þ YðtÞAT ðtÞ þ Q ðtÞ 0 ð49Þ  1 : Yðkhþ Þ ¼ YðkhÞ  YðkhÞCT ðkÞð R T C 0 ðkÞYðkhÞ; k A Z d ðkÞ þ C 0 ðkÞYðkhÞC 0 ðkÞÞ 0 khot r ðk þ 1Þh, where 

Q ðtÞ ¼ A1 ðtÞPc ðtÞAT1 ðtÞ þ BðtÞVBT ðtÞ; 

R d ðkÞ ¼ C 1 ðkÞPc ðkhÞC T1 ðkÞ

tAR

þ Dd ðkÞV d DTd ðkÞ:

ð50Þ

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Eq. (49) is a special case of the differential equation with finite jumps of Riccati type (28). Hence, the concept of stability solution of Eq. (49) is similar to the one introduced in connection with Eq. (28). Also, conditions which guarantee the existence and uniqueness of the stability and ~ θ-periodic solution Y s ðÞ of the differential equation with jumps of Riccati type (49) which satisfy the sign condition 

R d ðkÞ þ C 0 ðkÞY s ðkhÞCT0 ðkÞ40

for all k A Z;

ð51Þ

are obtained specializing the results from Propositions 6 and 7. Remark 4. If the assumptions (H1) and (H4) are fulfilled then the differential equation on S n X_ ðtÞ þ AT0 ðtÞXðtÞ þ XðtÞA0 ðtÞ þ AT1 ðtÞXðtÞ⋆ þ I n ¼ 0

ð52Þ



has a global solution X : R-S n which isbounded and uniform positive, that is, there exist constants μj 40, j¼ 1, 2 such that μ1 I n r X ðtÞr μ2 I n for all t A R (see [11,8]). So, Eq. (52) yields 

  d X ðtÞ þ AT0 ðt Þ X ðt Þ þ X ðt ÞA0 ðt Þ þ I n r 0: dt

This guarantees the exponential stability of the differential equation (32). Therefore, if the assumptions (H1) and (H4) are fulfilled, then the assumption (b) in Proposition 7 is also fulfilled.

Remark 5. If in Eq. (1) Aj ðtÞ ¼ Aj , j¼ 0,1, BðtÞ ¼ B, t A R then the unique bounded solution of the differential equation (48) is constant. Therefore Pc ðtÞ ¼ Pc , t A R and solves the Lyapunov type equation A0 Pc þ Pc AT0 þ A1 Pc AT1 þ BVBT ¼ 0:

ð53Þ

Under these conditions if the sequences fC j ðkÞgk A Z , j ¼ 0,1, fDd ðkÞgk A Z are periodic sequences with period θd Z 1, then the unique bounded and stabilizing solution Y s ðÞ of the differential equation with finite jumps of Riccati type (49) is a periodic function of period θ~ ¼ θd h. In the special case θd ¼ 1 one recovers the case considered in [7]. The next result provides the state-space representation of the optimal filter for the optimal filtering problem described in Section 2. Theorem 11. Assume that: (a) the assumptions (H1)–(H4) are fulfilled; (b) the differential equation with finite jumps of Riccati type (49) has a stabilizing and ~ θ-periodic solution Y s ðÞ satisfying the sign condition (51). 

Let G f be the filter having the state-space representation described by 8 x_ f ðtÞ ¼ A0 ðtÞxf ðtÞ > > <   xf ðkhþ Þ ¼ ðI n þ K s ðkÞC 0 ðkÞÞxf ðkhÞ K s ðkÞYðkhÞ > > : zf ðtÞ ¼ C z ðtÞxf ðtÞ

ð54Þ

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khot r ðk þ 1Þh, k ¼ 0; 1; …, where 



K s ðkÞ ¼  Y s ðkhÞC T0 ðkÞð R d ðkÞ þ C 0 ðkÞY s ðkhÞC T0 ðkÞÞ  1

ð55Þ



R d ðkÞ being introduced in Eq. (50). Under the considered assumptions the filter Gf lies in F ad and it is the solution for the optimization problem (5). More precisely, we have

Z ~ 
  1 θ  J Gf Z J G f ¼ Tr C z ðtÞY s ðtÞC Tz ðtÞ dt: ð56Þ ~θ 0

Proof. Under the assumptions (H1), (H2) and (H4) the differential equation (48) has a unique ~ So, the differential equation with finite jumps of solution Pc ðÞ which is periodic with period θ. Riccati type (49) is well defined. Based on the assumption (b) in the statement we can design the filter (54)–(55). This filter is of type (2) with the state space dimension nf. In this case the corresponding linear differential equation with finite jumpsof type (4) is given by Eq. (29), where Ks is defined in Eq. (55). Thus, we may conclude that G f described by Eq. (54) belongs to 

F ad . It remains to show that among all the admissible filters, G f achieves the smallest value of  the cost functional (3). Let Gf A F ad be arbitrary but fixed. Let Y ðÞ be the unique θ~ periodic solution of the corresponding equation of type (10). If ! Y 11 ðtÞ Y 12 ðtÞ  Y ðtÞ ¼ ⋆ Y 22 ðtÞ such that Y 11 ðtÞA S n , Y 22 A S nf we define ! Y 11 ðtÞ Y s ðtÞ Y 12 ðtÞ UðtÞ ¼ ⋆ Y 22 ðtÞ By direct calculations involving Eqs. (47)–(50) together with the identity Y 11 ðtÞ ¼ Pc ðtÞ, t A R we ~ obtain that t↦UðtÞ is a θperiodic solution of the differential equation with finite jumps on S nþnf 8 T < U_ ðtÞ ¼ A0cl ðtÞUðtÞ þ UðtÞA0cl ðtÞ; khot r ðk þ 1Þh  ⋎ ð57Þ : Uðkhþ Þ ¼ Ad0cl ðkÞUðkhÞATd0cl ðkÞ þ Bcl ðkÞ R d ðkÞC0 ðkÞY s ðkhÞCT0 ðkÞ ⋆; k A Z where

0

1  K ðkÞ s A: Bcl ðkÞ ¼ @  Bfd ðkÞ ⋎

If we take m þ nf , M 0 ðtÞ ¼ A0cl ðtÞ, M 1 ðtÞ ¼ 0, SðtÞ ¼ 0, t A R, M d0 ðkÞ ¼ Ad0cl ðkÞ, M d1 ðkÞ ¼ 0,  ⋎T ⋎ Sd ðkÞ ¼ Bcl ðkÞ R d ðkÞ þ C0 ðkÞY s ðkhÞCT0 ðkÞ Bcl ðkÞ, we remark that Eq. (57) is a special case of (17). In order to apply Proposition 3 in the case of Eq. (57) we have to show that the zero

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solution of the differential equation with finite jumps ( x_ cl ðtÞ ¼ A0cl ðtÞxcl ðtÞ dt; khot r ðk þ 1Þh xcl ðkhþ Þ ¼ Ad0cl ðkÞxcl ðkhÞ;

ð58Þ

kAZ

is exponentially stable. If ξcl ðÞ is a solution of Eq. (58) then the sequence fξcl ðkhÞgk A Z solves the discrete-time linear equation ξððk þ 1ÞhÞ ¼ MðkÞξðkhÞ

ð59Þ

where MðkÞ ¼ ϕ0cl ððk þ 1Þh; khÞAd0cl ðkÞ, ϕ0cl ðt; sÞ being the fundamental matrix solution of the linear differential equation x_ cl ðtÞ ¼ A0cl ðtÞxcl ðtÞ: The exponential stability of the zero solution of Eq. (58) is equivalent to the exponential stability of discrete-time linear equation (59). From Eq. (7) we obtain that ! 0 ϕ0 ðt; sÞ ϕ0cl ðt; sÞ ¼ 0 ϕf ðt; sÞ where ϕ0 ðt; sÞ is the fundamental matrix solution of x_ ðtÞ ¼ A0 ðtÞxðtÞ and ϕf ðt; sÞ is the fundamental matrix solution of x_ f ðtÞ ¼ Af ðtÞxf ðtÞ;

t A R:

It follows that MðkÞ ¼

ϕ0 ððk þ 1Þh; khÞ 0 ΞðkÞ ϕf ððk þ 1Þh; khÞ

!

where ΞðkÞ ¼ ϕf ððk þ 1Þh; khÞBfd ðkÞC 0 ðkÞ. Using Remark 4 together with the exponential stability of the linear differential equation with finite jumps (4) and the triangular structure of the matrix MðkÞ we may conclude that the zero solution of Eq. (59) is exponentially stable. Applying Proposition 3 in the case of Eq. (57) we deduce that UðtÞZ 0;

t AR:

ð60Þ

Further, we rewrite Eq. (44) in the form Z Z ~  
 1 θ~  1 θ  T J Gf ¼ Tr Cz ðtÞY s ðtÞCz ðtÞ dt þ Tr C cl ðtÞUðtÞC Tcl ðtÞ dt ~θ 0 ~θ 0 Combining Eqs. (60) and (61) we infer that Z
 1 θ~   J Gf Z Tr Cz ðtÞY s ðtÞC Tz ðtÞ dt θ~ 0

ð61Þ

ð62Þ 

for all Gf A F ad . It remains to show that in the special case of filter G f described by Eqs. (54)– (55) the inequality (62) becomes equality. Let us remark that in the special case of filter described by Eq. (54) we have C cl ðtÞUðtÞ⋆ ¼ C z ðtÞðI n  I n ÞUðtÞðI n  I n ÞT C Tz ðtÞ ¼ Cz U^ 11 ðtÞC Tz ðtÞ

ð63Þ

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where U^ 11 ðtÞ is the 11-block from ! ! U^ 11 ðtÞ U^ 12 ðtÞ In  In ¼ UðtÞ⋆ T 0 In U^ 12 ðtÞ U^ 22 ðtÞ ~ By direct calculations involving Eqs. (7), (54), (55) and (57) we obtain that t↦U^ 11 ðtÞ is a θperiodic solution of the following linear differential equations with finite jumps: 8 d > > < U^ 11 ðt Þ ¼ A0 ðt ÞU^ 11 ðt Þ þ U^ 11 ðt ÞAT0 ðt Þ dt ð64Þ   > þ > : U^ 11 ðkh Þ ¼ I n þ K s ðkÞC 0 ðkÞ U^ 11 ðkhÞ⋆ 

khot r ðk þ 1Þh. Since K s ðtÞ is a stabilizing gain we deduce via Proposition 3 that the ~ solution. Hence, U^ 11 ðtÞ ¼ 0, t A R. Plugging differential equation (64) has a unique θperiodic 

this equality in Eq. (63) we obtain that Eq. (62) reduces to Eq. (56) when Gf ¼ G f . Thus the proof is completed.□

Remark 6. In order to compute the coefficients of the optimal filter we need to know the values ^ Ys(jh), 0r j r θ^  1, of the stabilizing and θperiodic solution of the differential equation with finite jumps of Riccati type (49). The main steps in the computation of the coefficients of the optimal filter are: ~ (i) one computes the unique θperiodic solution of the differential equation (48) or the unique solution of the algebraic equation (53) if it is the case; (ii) one construct the discrete-time Riccati equation:  1 Yðk þ 1Þ ¼ AðkÞYðkÞ⋆  AðkÞYðkÞCT0 ðkÞ R d ðkÞ þ C0 ðkÞYðkÞCT0 ðkÞ ⋆ þ Sd ðkÞ ð65Þ where we denoted 8 < AðkÞ ¼ ϕ0 ððk þ 1Þh; khÞ  R ðkþ1Þh : Sd ðkÞ ¼ kh ϕ0 ððk þ 1Þh; sÞ Q ðsÞ⋆ ds 



ð66Þ

R d ðkÞ and Q ðsÞ are the ones defined in Eq. (50). ϕ0 ððk þ 1Þh; khÞ are the values of the fundamental matrix solution of the differential equation on Rn (32). ^ One computes the stabilizing and θperiodic solution Ys ðkÞ of Eqs. (65)–(66) satisfying the  T sign conditions C0 ðkÞYs ðkÞC 0 ðkÞ þ R d ðkÞ40, 0 r k r θ^  1; ^ (iii) one takes Y s ðjhÞ ¼ Ys ðjÞ, 0 r jr θ^  1, where Ys ðÞ is the unique stabilizing and θperiodic solution of the DTRE (65) (which is the discrete-time version of type (33) of the Riccati differential equation with finite jumps (49)–(50)).

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5. Numerical experiments To illustrate the proposed filtering strategy, we consider here a problem of field monitoring (see [15,16,22] and the reference therein) that will be adapted to our setting. In this problem, sensors are distributed on a rectangular region in order to estimate the state of a diffusion process described by the following partial differential equation: ∂ξðs; tÞ ¼ ∇2 ξðs; t Þ ∂t

ð67Þ

with Dirichlet boundary conditions: ξðs; Þ ¼ 0;

s A ∂D

ð68Þ

where ξðs; tÞ is the state value at location s and time t, ∇ denotes the Laplace operator, and ∂D denotes the boundary of the rectangular region of interest D. We consider a spatially discretized approximation of (67) (with boundary conditions (68)) which can be simply generated by setting [15,16,22]: 2

  ξði þ 1; j; tÞ  2ξði; j; tÞ þ ξði 1; j; tÞ ξði; j þ 1; tÞ 2ξði; j; tÞ þ ξði; j 1; tÞ ∇2 ξðs; tÞs ¼ ði;jÞ þ d2 d2

ð69Þ

where i ¼ 0; 1; …; lh , j ¼ 0; 1; …; lv , lh þ 2 and lv þ 2 are the width and length of the considered rectangular region, d is the physical distance between the lattice points and ξð  1; j; tÞ ¼ ξðlh þ 1; j; tÞ ¼ ξði;  1; tÞ ¼ ξði; lv þ 1; tÞ ¼ 0 for all position indices i; j and time t. From Eqs. (67)–(69), one can deduce easily the evolution equations: d xð t Þ ¼ A 0 xð t Þ dt where xðtÞA Rnx , nx ¼ ðlh þ 1Þ  ðlv þ 1Þ, denotes the state vector xðtÞ ¼ ½ξð0; 0; tÞ ξð0; 1; tÞ ⋯ ξðlh ; lv ; tÞT and A0 can be directly computed from Eq. (69). In order to fit in with our setting, we introduce a process noise and a state-dependent noise yielding dxðtÞ ¼ A0 xðtÞ dt þ A1 xðtÞ dwðtÞ þ B dvðtÞ where A1 and B have been randomly generated. We assume that M sensors, Monx , are deployed to make measurements of the state. In our example, we choose lh ¼ lv ¼ 4 (hence nx ¼ 25) and M ¼ 10. The location of the sensors as well as the state variables in the rectangular field are illustrated in Fig. 1. We consider that the sensors and the filter communicate over a communication channel. We assume that the communication channel induces some communication constraints. More specifically, the constraint we deal with here is referred to in the literature as a medium access constraint. In this case, the shared network can only accommodate a limited number of simultaneous communications between components. In this context, it is only meaningful to specify a filter in conjunction with a communication policy which indicates the times at which the plant's sensors are to be granted medium access. This communication policy is known in the literature as communication sequence. The communication sequence specifies which sensors are able to send information to the filter at each time step. To meet this requirement, we use a periodic communication sequence [2,12] which can be described as follows. Suppose that there are q sensor nodes which are indexed from 1 to q, each node contains r sensors. Due to communication limitation, at each sampling time, only one sensor node is allowed to send the measurements to the filter. The sensor nodes communicate with the plant

V. Dragan et al. / Journal of the Franklin Institute 352 (2015) 5985–6010

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5

4

2

Y−axis (m)

3

2

8

4

6

1

1

7

9

3

0

−1 −1

10

5

0

1

2

3

4

5

X−axis (m)

Fig. 1. M¼ 10 sensors deployed in a 6  6 region: n—boundary points, —estimated points,

—sensor locations.

following a periodic pattern with period θd . For simulation purposes, we fixed q ¼ 5, r ¼ 4 and θd ¼ 5. The different sensor nodes are given as follows: 8 0 1 0 1 0 1 0 1 0 19 3 5 7 1 9 > > > > > B C B C B C B C B C> = < B 2 C B 4 C B 6 C B 8 C B 10 C Nodes ¼ B C; B C; B C; B C; B C > @ 3 A @ 5 A @ 7 A @ 9 A @ 1 A> > > > > ; : 6 8 10 4 2 i where kj represents the node containing the corresponding sensors i; j; k; l in Fig. 1. The l

θd -periodic communication sequence sk ¼ skþθd , k Z 0, is given as follows:   fsk gk Z 0 ¼ Nodesð1Þ; Nodesð2Þ; Nodesð3Þ; Nodesð4Þ; Nodesð5Þ; Nodesð1Þ; ⋯ where NodesðiÞ represents the i-th element of the set Nodes. Hence, the measurements of the state are done according to yðkhÞ ¼ C 0 ðkÞxðkhÞ where h40 is the sampling period and the matrices C0 ðk þ θd Þ ¼ C0 ðkÞ, k Z 0 are the measurements matrices corresponding to the communication sequence fsk gk Z 0 . In order to fit in with our setting, we introduce a measurement noise and a state-dependent noise leading to yðkhÞ ¼ ½C0 ðkÞ þ C 1 ðkÞwd ðkÞxðkhÞ þ Dd ðkÞvd ðkÞ where C1 ðk þ θd Þ ¼ C1 ðkÞ, Dd ðk þ θd Þ ¼ Dd ðkÞ, 8 k Z 0. Note that, in our experiments, the matrices C 1 ðiÞ and Dd(i), 0 r ir θd  1, are generated randomly. Fig. 2 shows the evolution of the estimation error eðtÞ ¼ xðtÞ  xf ðtÞ (due to the large number of state variables, we have only showed here the evolution of some components of e(t)). We have also performed Monte Carlo simulations (corresponding to different realizations of the stochastic processes). We have performed 1000 simulations for the estimation error system. Fig. 3 shows the average estimation error obtained from the 1000 realizations (here also, we have only showed

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V. Dragan et al. / Journal of the Franklin Institute 352 (2015) 5985–6010 20

40 x(6,t)−xf(6,t) x(10,t)−xf(10,t)

30

0

Estimation error

Estimation error

x(1,t)−xf(1,t) x(3,t)−xf(3,t)

−20

−40

20 10 0 −10

−60

0

2

4

6

8

−20

10

0

2

4

Time 20

10

x(23,t)−xf(23,t) x(25,t)−xf(25,t)

40 Estimation error

Estimation error

8

60 x(15,t)−xf(15,t) x(21,t)−xf(21,t)

10 0 −10 −20 −30

6 Time

20 0 −20

0

2

4

6

8

−40

10

0

2

4

Time

6

8

10

Time

Fig. 2. Time responses of the estimation errors: sample path realization of the stochastic processes. 40 Aver(x(2,t)−xf(2,t)) Aver(x(4,t)−xf(4,t))

10

Average estimation error

Average estimation error

15

5 0 −5 −10 −15

0

2

4

6

8

20 10 0 −10 −20

10

Aver(x(8,t)−xf(8,t)) Aver(x(12,t)−xf(12,t))

30

0

2

Time (sec)

6

8

10

10 Aver(x(14,t)−xf(14,t)) Aver(x(19,t)−xf(19,t))

4

Average estimation error

Average estimation error

6

2 0 −2 −4 −6 −8

4

Time (sec)

0

2

4

6

8

10

Aver(x(20,t)−xf(20,t)) Aver(x(24,t)−xf(24,t))

5 0 −5 −10 −15 −20

0

Time (sec)

2

4

6

Time (sec)

Fig. 3. Average estimation errors.

8

10

V. Dragan et al. / Journal of the Franklin Institute 352 (2015) 5985–6010

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the
evolution of someP components of the average estimation error vector) where 1000 1 Aver xðt Þ xf ðt Þ ¼ 1000 j ¼ 1 ðxðtÞ xf ðtÞÞj and j represents the stochastic processes realizati on index.

6. Conclusion In this paper, we have considered the problem of optimal H2 filtering for a class of continuous-time periodic stochastic systems with sampled measurements. The optimal filtering problem has been formulated and solved in a continuous-time systems with jumps setting. The solution of the considered optimization problem has been expressed in terms of the stabilizing solution of a suitable generalized Riccati differential equation. The numerical implementation of the obtained filter requires the use of methods which are quite standard.

Acknowledgments This work was supported by CNCS Romania [Grant number 145/2011]. References [1] S. Bittanti, P. Colaneri, Periodic Systems, Filtering and Control, Springer-Verlag, London, 2009. [2] R.W. Brockett, Stabilization of motor networks, in: Proceedings of the 34th IEEE Conference on Decision & Control, 1995, pp. 1484–1488. [3] O.L.V. Costa, G.R.A.M. Benites, Linear minimum mean square filter for discrete-time linear systems with Markov jumps and multiplicative noises, Autom. J. IFAC 47 (2011) 466–476. [4] Ju.L. Daleckiı̆, M.G. Kreı̆n, Stability of solutions of differential equations in Banach space, Translations of Mathematical Monographs, vol. 43, American Mathematical Society, Providence, Rhode Island, USA, 1974. [5] V. Dragan, T. Damm, G. Freiling, T. Morozan, Differential equations with positive evolutions and some applications, Results Math. 48 (2005) 206–236. [6] V. Dragan, T. Morozan, A.-M. Stoica, Mathematical Methods in Robust Control of Discrete-time Linear Stochastic Systems, Springer, New York, USA, 2010. [7] V. Dragan, A.-M. Stoica, Optimal H2 filtering for a class of linear stochastic systems with sampling, Autom. J. IFAC 48 (2012) 2494–2501. [8] V. Dragan, T. Morozan, A.-M. Stoica, Mathematical Methods in Robust Control of Linear Stochastic Systems, second ed., Springer, New York, USA, 2013. [9] V. Dragan, S. Aberkane, H2 optimal filtering for continuous-time periodic linear stochastic systems with statedependent noise, Syst. Control Lett. 66 (2014) 35–42. [10] E. Gershon, U. Shaked, I. Yaesh, H 1 Control and Estimation of State-Multiplicative Linear Systems, Lecture Notes in Control and Information Sciences, vol. 318, Springer-Verlag London Limited, London, England, 2005. [11] R.Z. Hasminskii, Stochastic Stability of Differential Equations, Sijthoff & Noordhoff, Alphen aan den Rijn, 1980. [12] D. Hristu, K. Morgansen, Limited communication control, Syst. Control Lett. 37 (1999) 193–205. [13] A. Ichikawa, H. Katayama, Linear Time Varying Systems and Sampled-data Systems, Lecture Notes in Control and Information Sciences, vol. 265, Springer-Verlag, London, 2001. [14] A.H. Jazwinski, Stochastic Processes and Filtering Theory, Mathematics in Science and Engineering, 1970. [15] S. Liu, M. Fardad, E. Masazade, P. Varshney, Optimal periodic sensor scheduling in large-scale dynamical networks, IEEE Trans. Signal Process. 62 (12) (2014) 3055–3068. [16] Y. Mo, R. Ambrosino, B. Sinopoli, Sensor selection strategies for state estimation in energy constrained wireless sensor networks, Autom. J. IFAC 47 (7) (2011) 1330–1338. [17] B. Oksendal, Stochastic Differential Equations, Springer, Berlin Heidelberg, Germany, 1998. [18] M. Souza, A.R. Fioravanti, J.C. Geromel, H2 sampled-data filtering of linear systems, IEEE Trans. Signal Process. 62 (18) (2014) 4839–4846.

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[19] A.M. Stoica, I. Yaesh, Kalman-type filtering for discrete-time stochastic systems with state-dependent noise, in: Proceedings of the 18th International Symposium on Mathematical Theory of Networks and Systems, Blacksburg, Virginia, USA, 2008. [20] A.M. Stoica, V. Dragan, I. Yaesh, Kalman-type filtering for stochastic systems with state-dependent noise and Markovian jumps, in: Proceedings of the 15th IFAC Symposium on System Identification, Saint-Malo, France, 2009. [21] A. Varga, On solving periodic Riccati equations, Numer. Linear Algebra Appl. 15 (9) (2008) 809–835. [22] H. Zhang, J. Moura, B. Krogh, Dynamic field estimation using wireless sensor networks: tradeoffs between estimation error and communication cost, IEEE Trans. Signal Process. 57 (6) (2009) 2383–2395.