2 Observers Design for a Class of Lipschitz Discrete-Time Systems with Time-Delay Ali Zemouche and Mohamed Boutayeb Centre de Recherche en Automatique de Nancy, CRAN UMR 7039 CNRS, Nancy-Université, 54400 Cosnes et Romain France Introduction The observer design problem for nonlinear time-delay systems becomes more and more a subject of research in constant evolution Germani et al (2002), Germani & Pepe (2004), Aggoune et al (1999), Raff & Allgöwer (2006), Trinh et al (2004), Xu et al (2004), Zemouche et al (2006), Zemouche et al (2007) Indeed, time-delay is frequently encountered in various practical systems, such as chemical engineering systems, neural networks and population dynamic model One of the recent application of time-delay is the synchronization and information recovery in chaotic communication systems Cherrier et al (2005) In fact, the time-delay is added in a suitable way to the chaotic system in the goal to increase the complexity of the chaotic behavior and then to enhance the security of communication systems On the other hand, contrary to nonlinear continuous-time systems, little attention has been paid toward discrete-time nonlinear systems with time-delay We refer the readers to the few existing references Lu & Ho (2004a) and Lu & Ho (2004b), where the authors investigated the problem of robust H∞ observer design for a class of Lipschitz time-delay systems with uncertain parameters in the discrete-time case Their method show the stability of the state of the system and the estimation error simultaneously This chapter deals with observer design for a class of Lipschitz nonlinear discrete-time systems with time-delay The main result lies in the use of a new structure of the proposed observer inspired from Fan & Arcak (2003) Using a Lyapunov-Krasovskii functional, a new nonrestrictive synthesis condition is obtained This condition, expressed in term of LMI, contains more degree of freedom than those proposed by the approaches available in literature Indeed, these last use a simple Luenberger observer which can be derived from the general form of the observer proposed in this paper by neglecting some observer gains An extension of the presented result to H∞ performance analysis is given in the goal to take into account the noise which affects the considered system A more general LMI is established The last section is devoted to systems with differentiable nonlinearities In this case, based on the use of the Differential Mean Value Theorem (DMVT), less restrictive synthesis conditions are proposed Notations : The following notations will be used throughout this chapter • is the usual Euclidean norm; 20 Discrete Time Systems • ( ) is used for the blocks induced by symmetry; • A T represents the transposed matrix of A; • Ir represents the identity matrix of dimension r; • for a square matrix S, S > (S < 0) means that this matrix is positive definite (negative definite); • zt (k) represents the vector x (k − t) for all z; • The notation x s = ∑ ∞ x (k) k= 2 is the s norm of the vector x ∈ R s The set s is defined by s = x ∈ Rs : x s < +∞ Problem formulation and observer synthesis In this section, we introduce the class of nonlinear systems to be studied, the proposed state observer and the observer synthesis conditions 2.1 Problem formulation Consider the class of systems described in a detailed forme by the following equations : x (k + 1) = Ax (k) + Ad xd (k) + B f Hx (k), Hd xd (k) (1a) y(k) = Cx (k) (1b) x (k) = x (k), for k = − d, , (1c) where the constant matrices A, Ad , B, C, H and Hd are of appropriate dimensions The function f : R s1 × R s2 → R q satisfies the Lipschitz condition with Lipschitz constant γ f , i.e : ˆ z1 − z1 ˆ ˆ ˆ ˆ , ∀ z1 , z2 , z1 , z2 ≤ γf (2) f z1 , z2 − f z1 , z2 ˆ z2 − z2 Now, consider the following new structure of the proposed observer defined by the equations (78) : ˆ ˆ ˆ x ( k + 1) = A x ( k ) + A d x d ( k ) + B f v ( k ), w ( k ) ˆ ˆ + L y(k) − C x(k) + L d yd (k) − C xd (k) ˆ ˆ ˆ v(k) = H x (k) + K y(k) − C x (k) + K d yd (k) − C xd (k) ˆ ˆ ˆ w(k) = Hd xd (k) + K y(k) − C x(k) + Kd yd (k) − C xd (k) (3a) (3b) (3c) Observers Design for a Class of Lipschitz Discrete-Time Systems with Time-Delay 21 The dynamic of the estimation error is : ε(k + 1) = A − LC ε(k) + Ad − L d C ε d (k) + Bδ f k with (4) δ f k = f Hx (k), Hd xd (k) − f v(k), w(k) From (35), we obtain ( H − K C )ε(k) − Kd Cε d (k) C )ε (k ) − K Cε (k ) ( Hd − Kd d δ fk ≤ γ f (5) 2.2 Observer synthesis conditions This subsection is devoted to the observer synthesis method that provides a sufficient condition ensuring the asymptotic convergence of the estimation error towards zero The synthesis conditions, expressed in term of LMI, are given in the following theorem Theorem 2.1 The estimation error is asymptotically stable if there exist a scalar α > and matrices ¯ ¯ ¯1 ¯2 P = P T > 0, Q = Q T > 0, R, Rd , K , K , Kd and Kd of appropriate dimensions such that the following LMI is feasible : ⎤ ⎡ T T − P + Q M13 M14 M15 M 16 ⎥ ⎢ ⎥ ⎢ T T ⎢ ( ) −Q M M M 25 M 26 ⎥ 23 24 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ( ) ( ) M33 0 ⎥ ⎢ ⎥ ⎢ (6) ⎥ is the disturbance attenuation level to be minimized under some conditions that we will determined later The inequality (17) is equivalent to ε s λ ≤ √ ω s + ωd s − −1 ∑ k =− d ω (k) (18) Without loss of generality, we assume that ω (k) = for k = − d, , −1 Then, (18) becomes ε s λ ≤ √ ω s + ωd 2 s (19) Remark 3.1 In fact, if ω (k) = for k = − d, , −1, we must replace the inequality (17) by ε s ≤λ ω s + −1 ∑ ω (k) k=−d (20) in order to obtain (19) Robust H∞ observer design problem Li & Fu (1997) : Given the system (14) and the observer (15), then the problem of robust H∞ observer design is to determine the matrices L, L d , K , K , Kd and Kd so that lim ε(k) = for ω (k) = 0; k→∞ ε s ≤λ ω s ∀ ω (k) = 0; ε(k) = 0, k = − d, , (21) (22) From the equivalence between (17) and (19), the problem of robust H∞ observer design (see the Appendix) is reduced to find a Lyapunov function Vk such that Wk = ΔV + ε T ( k) ε ( k) − where λ2 T λ2 T ω ( k) ω ( k) − ω ( k) ωd ( k) < 2 d (23) ΔV = Vk+1 − Vk At this stage, we can state the following theorem, which provides a sufficient condition ensuring (23) 26 Discrete Time Systems Theorem 3.2 The robust H∞ observer design problem corresponding to the system (14) and the observer (15) is solvable if there exist a scalar α > matrices P = P T > 0, Q = Q T > 0, ¯ ¯ ¯1 ¯2 R, Rd , K , K , Kd and Kd of appropriate dimensions so that the following convex optimization problem is feasible : min(γ ) subject to Γ < (24) where ⎡⎡ ⎤ − P + Q + In M13 0 ⎢⎢ ( ) − Q M23 0 ⎥ ⎢⎢ ⎥ ⎢⎢ ( ) ( ) M33 M34 M35 ⎥ ⎢⎢ ⎥ ⎢⎣ ( ) ( ) ( ) − γIs ⎦ ⎢ ⎢ ( ) ( ) ( ) ( ) − γIs ⎢ ⎢ Γ=⎢ ⎢ ⎡ T T ⎤T ⎢ M 14 M 15 M 16 ⎢ T T MT ⎥ ⎢ ⎢ M 24 M 25 26 ⎥ ⎢ ⎢ ⎢ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎣ET P − C T R ⎣ ⎦ ω − Dω R d 0 T T ⎤⎤ M 14 M 15 M 16 T T T ⎢ M 24 M 25 M 26 ⎥⎥ ⎢ ⎥⎥ ⎢ 0 ⎥⎥ ⎢ ⎥⎥ ⎣E T P − C T R 0 ⎦⎥ ⎥ ω − Dω R d 0 ⎥ ⎥ ⎥ ⎥ ⎥ ⎡ ⎤⎥ ⎥ −P 0 ⎥ ⎢ ( ) − αγ2 Is ⎥⎥ ⎣ ⎦⎥ f ⎥ ( ) ( ) − αγ2 Is2 ⎦ f ⎡ (25) with M 34 = B T PEω − B T R T C, M 35 = − B T (26a) T R d Dω , (26b) and M 13 , M 14 , M 15 , M 16 , M 24 , M 25 , M 26 , M 33 are de ned in (7) The gains L and L d , K , K , Kd , Kd and the minimum disturbance attenuation level λ are given respectively by T L = P −1 R T , L d = P −1 R d ¯1 ¯ K , K2 = K2, α α ¯1 ¯2 Kd = Kd , Kd = Kd , α α K1 = λ= 2γ Proof The proof of this theorem is an extension of that of Theorem 2.1 Let us consider the same Lyapunov-Krasovskii functional defined in (8) We show that if the convex optimization problem (24) is solvable, we have Wk < Using the dynamics (16), we obtain (27) Wk = η T S1 η where ⎡ ⎡ ⎤ ⎤ ⎡ T ˜ ˜ ˜ A P E˜ω − A T P Dω In 0 ⎢ ˜ ˜ ⎣ A T P E˜ω − A T P Dω ⎦ ˜ M + ⎣ 0 0⎦ ⎢ d d ⎢ T PE −BT PD ˜ω ˜ω 00 B ⎢ S1 = ⎢⎡ T ⎤ ˜ ˜ ˜ ⎢ A P E˜ω − A T P Dω T ˜T ˜ ˜T ⎢ Eω P D ω Eω P E˜ω − γIs ˜ ˜ ˜ ⎣⎣ A T P E˜ω − A T P Dω ⎦ d d ˜T ˜T ˜ Dω P E˜ω Dω P Dω − γIs ˜ B T P E˜ω − B T P Dω ⎤ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦ (28) Observers Design for a Class of Lipschitz Discrete-Time Systems with Time-Delay 27 where E˜ω = Eω − LC ˜ Dω = L d Dω (29b) T η T = εT εT δ f k ω T ωd , d (29c) γ= λ2 (29a) (29d) ˜ ˜ The matrices M , A and Ad are defined in (9) As in the proof of Theorem 2.1, since δ f k satisfies (5), we deduce, after multiplying by a scalar α > 0, that (30) η T S2 η ≥ where ⎡ ⎢ ⎢ S2 = ⎢ ⎣ and M is defined in (11b) The inequality (31) implies that M3 αγ2 f 0 0 00 ⎤ ⎥ − αIq 0⎥ ⎥ 0 0⎦ 00 (31) Wk = η T (S1 + S2 )η (32) Now, using the Schur Lemma and the notations R = and Rd = we deduce that the inequality S1 + S2 < is equivalent to Γ < The estimation error converges robustly √ asymptotically to zero with a minimum value of the disturbance attenuation level λ = 2γ if the convex optimization problem (24) is solvable This ends the proof of Theorem 3.2 LT P T L d P, Remark 3.3 We can obtain a synthesis condition which contains more degree of freedom than the LMI (6) by using a more general design of the observer This new design of the observer can take the following structure : ˆ ˆ ˆ x ( k + 1) = A x ( k ) + A d x d ( k ) + B f v ( k ), w ( k ) ˆ + L y(k) − C x (k) + d ∑ Li i =1 ˆ ˆ v(k) = H x (k) + K y(k) − C x (k) + ˆ yi (k) − C xi (k) d ∑ Ki1 i =1 ˆ ˆ w(k) = Hd xd (k) + K y(k) − C x(k) + d ∑ Ki2 i =1 ˆ yi (k) − C xi (k) ˆ yi (k) − C xi (k) (33a) (33b) (33c) 28 Discrete Time Systems If such an observer is used, the adequate Lyapunov-Krasovskii functional that we propose is under the following form : Vk = ε T (k) Pε(k) + j=d i= j ∑∑ j =1 i =1 ε T (k) Q j ε i (k) i (34) Systems with differentiable nonlinearities 4.1 Reformulation of the problem In this section, we need to assume that the function f is differentiable with respect to x Rewrite also f under the detailed form : ⎡ d ⎤ f ( H1 x, H1 z) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ f ( Hx, Hd z) = ⎢ ⎥ ⎣ ⎦ d f q ( Hq x, Hq z) (35) where Hi ∈ R s i ×n and Hid ∈ R r i ×n for all i ∈ {1, , q } Here, we use the following reformulation of the Lipschitz condition : − ∞ < aij ≤ ∂ fi i i (ζ , z ) ≤ bij < + ∞, ∀ ζ i ∈ R s i , ∀ zi ∈ R r i ∂ζ i j (36) − ∞ < ad ≤ ij ∂ fi i i d ( x , ζ ) ≤ bij < + ∞, ∀ ζ i ∈ R r i , ∀ x i ∈ R s i ∂ζ i j (37) where x i = Hi x and zi = Hid z The conditions (36)-(37) imply that the differentiable function f is γ f -Lipschitz where γf = i=q ⎛ j=si ∑ max ⎝ ∑ max i =1 j =1 | aij |2 , | bij |2 , j =r i ∑ max j =1 ⎞ d | ad |2 , | bij |2 ij ⎠ The reformulation of the Lipschitz condition for differentiable functions as in (36) and (37) plays an important role on the feasibility of the synthesis conditions and avoids high gain as shown in Zemouche et al (2008) In addition, it is shown in Alessandri (2004) that the use of the classical Lipschitz property leads to restrictive synthesis conditions Remark 4.1 For simplicity of the presentation, we assume, without loss of generality, that f satis es (36) and (37) with aij = and ad = for all i, l = 1, , q, j = 1, , s and m = 1, , r, where lm d s = max (si ) and r = max (ri ) Indeed, if there exist subsets S1 , S1 ⊂ {1, , q }, S2 ⊂ {1, , s} and 1≤ i ≤ q 1≤ i ≤ q d d d S2 ⊂ {1, , r } such that aij = for all (i, j) ∈ S1 × S2 and ad = for all (l, m) ∈ S1 × S2 , we can lm 34 Discrete Time Systems that we can write under the form (35) with d H1 = 0 , H1 = 0 d H2 = 0 , H2 = σ 0 Assume that the first component of the state x is measured, i.e : C = 0 The system exhibits a chaotic behavior for the following numerical values : α = 9, β = 14, γ = 5, d = δ = 5, = 1000, σ = 100 as can be shown in the figure The bounds of the partial derivatives of f are 100 x3 50 −50 −100 10 x2 −10 −5 x1 Fig Phase plot of the system d d a11 = 1, b11 = 1, a21 = −1, b21 = According to the remark 4.1, we must solve the LMI (48) with ⎤ ⎡ 0 d d ˜d ˜ b21 = b21 − a21 = 2, Ad = ⎣0 0 ⎦ 0 −T σ Hence, we obtain the following solutions : ⎤ ⎡ ⎤ ⎡ 1.3394 d d L = ⎣ 4.9503 ⎦ , L d = ⎣ ⎦ , K1 = 0.9999, K2 = −0.0425, K1 = −1.792 × 10−13 , K2 = 100 −1000 40.8525 The simulation results are shown in figure 35 Observers Design for a Class of Lipschitz Discrete-Time Systems with Time-Delay x2 0.8 x 0.6 the estimate of x1 0.4 0.2 Magnitude Magnitude the estimate of x −0.2 −0.4 −1 −2 −0.6 −3 −0.8 −1 50 100 −4 150 50 ˆ (a) The first component x1 and its estimate x1 100 150 Time (k) Tim (k) ˆ (b) The second component x2 and its estimate x2 30 20 Magnitude 10 −10 −20 x3 −30 the estimate of x −40 50 100 150 Tim (k) ˆ (c) The third component x3 and its estimate x3 Fig Estimation error behavior Conclusion This chapter investigates the problem of observer design for a class of Lipschitz nonlinear time-delay systems in the discrete-time case A new observer synthesis method is proposed, which leads to a less restrictive synthesis condition Indeed, the obtained synthesis condition, expressed in term of LMI, contains more degree of freedom because of the general structure of the proposed observer In order to take into account the noise (if it exists) which affects the considered system, a section is devoted to the study of H∞ robustness A dilated LMI condition is established particularly for systems with differentiable nonlinearities Numerical examples are given in order to show the effectiveness of the proposed results A Schur Lemma In this section, we recall the Schur lemma and how it is used in the proof of Theorem 2.1 36 Discrete Time Systems Lemma A.1 Boyd et al (1994) Let Q1 , Q2 and Q3 be three matrices of appropriate dimensions such T T that Q1 = Q1 and Q3 = Q3 Then, the two following inequalities are equivalent : Q1 Q2 < 0, T Q2 Q3 (80) − T Q3 < and Q1 − Q2 Q3 Q2 < (81) Now, we use the Lemma A.1 to demonstrate the equivalence between M + M < and M < We have ⎡ ⎤ ⎡ T ˜ ˜ ˜ ˜ ˜ −P + Q A T PB A P A A T P Ad ˜T ˜T ˜ M1 + M2 = ⎣ ( ) − Q A d PB ⎦ + ⎣ ( ) A d P A d ( ) ( ) ( ) ( ) B T PB − αIq ⎡ T T T T M 15 M 15 + M 16 M 16 M 15 M 25 + M 16 M 26 ⎣ T T + ( ) M 26 M 26 + M 25 M 25 αγ2 f ( ) ( ) By isolating the matrix M1 + M2 = ⎣ (82) ⎡ ⎤ P 0 ⎢ ⎥ Λ = ⎣ αγ f Is1 ⎦ 0 αγ2 Is2 f we obtain ⎡ ⎤ 0⎦ ⎤ 0⎦ ⎤ ⎡ T T ˜ A T M 15 M 16 ⎤ ⎡ ˜ A ˜ Ad ⎢ ⎥ ⎢ −P + Q ⎢ ⎥ ⎢ T T ˜T ˜T ( ) −Q A d PB ⎦ − ⎢ A d M 25 M 26 ⎥ Υ(− Λ ) −1 Υ ⎢M 15 M 25 ⎥ ⎢ ⎢ T PB − αI ⎦ ⎣ ⎣ ( ) ( ) B q M 16 M 26 0 ˜ A T PB ⎤ ⎥ ⎥ 0⎥ ⎥ ⎦ (83) ⎡ ⎤ P 0 Υ = ⎣ Is ⎦ 0 Is where By setting ⎡ ˜T T T ⎤ A M 15 M 16 ˜ ⎥ ⎢ −P + Q A T PB ⎥ ⎢ ˜ T PB ⎦ , Q2 = ⎢ A T M T M T ⎥ Υ and Q3 = − Λ ˜ ⎣ ( ) −Q Q1 = Ad ⎢ d 25 26 ⎥ ⎦ ⎣ ( ) ( ) B T PB − αIq 0 ⎡ we have ⎤ − T M + M = Q1 − Q2 Q3 Q2 Since Q3 < 0, we deduce from the Lemma A.1 that M1 + M2 < (84) 37 Observers Design for a Class of Lipschitz Discrete-Time Systems with Time-Delay is equivalent to (80), which is equivalent to M4 < where M is defined in (13) This ends the proof of equivalence between M + M < and M < The Lemma A.1 is used of the same manner in theorem 3.2 B Some Details on Robust H∞ Observer Design Problem Hereafter, we show why the problem of robust H∞ observer design is reduced to find a Lyapunov function Vk so that Wk < 0, where Wk is defined in (23) In other words, we show that Wk < implies that the inequalities (21) and (22) are satisfied If ω (k) = 0, we have Wk < implies that ΔV < Then, from the Lyapunov theory, we deduce that the estimation error converges asymptotically towards zero, and then we have (21) Now, if ω (k) = 0; ε(k) = 0, k = − d, , 0, we obtain Wk < implies that N ∑ k =0 ε ( k) < λ2 N ∑ ω ( k) k =0 + λ2 N ∑ k =0 ωd ( k) − N ∑ (Vk +1 − Vk ) (85) k =0 Since without loss of generality, we have assumed that ω (k) = for k = − d, , −1 and ε(k) = 0, k = − d, , 0, we deduce that N ∑ k =0 ε ( k) < λ2 N ∑ k =0 ω ( k) + λ2 N −d ∑ k =0 ω ( k) − VN < λ2 N ∑ ω ( k) k =0 + λ2 N −d ∑ k =0 ω ( k) (86) When N tends toward infinity, we obtain ∞ ∑ k =0 ε ( k) As ≤ λ2 ∞ ∑ k =0 ω ( k) ∞ ∑ k =0 + λ2 ω (k) ∞−d ∑ k =0 = ω ( k) ∞−d ∑ k =0 ≤ ω (k) λ2 2 N ∑ k =0 ω ( k) = ω + λ2 N −d ∑ k =0 ω ( k ) (87) s then the final relation (22) is inferred C References Aggoune, W., Boutayeb, M & Darouach, M (1999) Observers design for a class of nonlinear systems with time-varying delay, CDC’1999, Phoenix, Arizona USA December Alessandri, A (2004) Design of observers for Lipschitz nonlinear systems using LMI, NOLCOS, IFAC Symposium on Nonlinear Control Systems, Stuttgart, Germany Boyd, S., El Ghaoui, L., Feron, E & Balakrishnan, V (1994) Linear matrix inequalities in system and control theory, SIAM Studies in Applied Mathematics, Philadelphia, USA Cherrier, E., Boutayeb, M & Ragot, J (2005) Observers based synchronization and input recovery for a class of chaotic models, Proceedings of the 44th IEEE Conference on Decision and Control and European Control Conference , Seville, Spain Cherrier, E., Boutayeb, M & Ragot, J (2006) Observers-based synchronization and input recovery for a class of nonlinear chaotic models, IEEE Trans Circuits Syst I 53(9): 1977–1988 Fan, X & Arcak, M (2003) Observer design for systems with multivariable monotone nonlinearities, Systems and Control Letters 50: 319–330 38 Discrete Time Systems Germani, A., Manes, C & Pepe, P (2002) A new approach to state observation of nonlinear systems with delayed output, IEEE Trans Autom Control 47(1): 96–101 Germani, A & Pepe, P (2004) An observer for a class of nonlinear systems with multiple discrete and distributed time delays, 16th MTNS, Leuven, Belgium Li, H & Fu, M (1997) A linear matrix inequality approach to robust H∞ filtering, IEEE Trans on Signal Processing 45(9): 2338–2350 Lu, G & Ho, D W C (2004a) Robust H∞ observer for a class of nonlinear discrete systems, Proceedings of the 5th Asian Control Conference, ASCC2004, Melbourne, Australia Lu, G & Ho, D W C (2004b) Robust H∞ observer for a class of nonlinear discrete systems with time delay and parameter uncertainties, IEE Control Theory Application 151(4) Raff, T & Allgöwer, F (2006) An EKF-based observer for nonlinear time-delay systems, 2006 American Control Conference ACC’06, Minneapolis, Minnesota, USA Trinh, H., Aldeen, M & Nahavandi, S (2004) An observer design procedure for a class of nonlinear time-delay systems, Computers & Electrical Engineering 30: 61–71 Xu, S., Lu, J., Zhou, S & Yang, C (2004) Design of observers for a class of discrete-time uncertain nonlinear systems with time delay, Journal of the Franklin Institute 341: 295–308 Zemouche, A., Boutayeb, M & Bara, G I (2006) On observers design for nonlinear time-delay systems, 2006 American Control Conference ACC’06, Minneapolis, Minnesota, USA Zemouche, A., Boutayeb, M & Bara, G I (2007) Observer design for a class of nonlinear time-delay systems, 2007 American Control Conference ACC’07, New York, USA Zemouche, A., Boutayeb, M & Bara, G I (2008) Observers for a class of Lipschitz systems with extension to H∞ performance analysis, Systems & Control Letters 57(1): 18–27 Distributed Fusion Prediction for Mixed Continuous-Discrete Linear Systems 1School Ha-ryong Song1, Moon-gu Jeon1 and Vladimir Shin2 of Information and Communications, Gwangju Institute of Science and Technology Department of Information statistics, Gyeong sang National University South Korea Introduction The integration of information from a combination of different types of observed instruments (sensors) are often used in the design of high-accuracy control systems Typical applications that benefit from this use of multiple sensors include industrial tasks, military commands, mobile robot navigation, multi-target tracking, and aircraft navigation (see (hall, 1992, Bar-Shalom, 1990, Bar-Shalom & Li, 1995, Zhu, 2002, Ren & Key, 1989) and references therein) One problem that arises from the use of multiple sensors is that if all local sensors observe the same target, the question then becomes how to effectively combine the corresponding local estimates Several distributed fusion architectures have been discussed in (Alouani, 2005, Bar-Shalom & Campo, 1986, Bar-Shalom, 2006, Li et al., 2003, Berg & Durrant-Whyte, 1994, Hamshemipour et al., 1998) and algorithms for distributed estimation fusion have been developed in (Bar-Shalom & Campo, 1986, Chang et al., 1997, Chang et al, 2002, Deng et al., 2005, Sun, 2004, Zhou et al., 2006, Zhu et al., 1999, Zhu et al., 2001, Roecker & McGillem, 1998, Shin et al, 2006) To this end, the Bar-Shalom and Campo fusion formula (Bar-Shalom & Campo, 1986) for two-sensor systems has been generalized for an arbitrary number of sensors in (Deng et al., 2005, Sun, 2004, Shin et al., 2007) The formula represents an optimal mean-square linear combination of the local estimates with matrix weights The analogous formula for weighting an arbitrary number of local estimates using scalar weights has been proposed in (Shin et al., 2007, Sun & Deng, 2005, Lee & Shin 2007) However, because of lack of prior information, in general, the distributed filtering using the fusion formula is globally suboptimal compared with optimal centralized filtering (Chang et al., 1997) Nevertheless, in this case it has advantages of lower computational requirements, efficient communication costs, parallel implementation, and fault-tolerance (Chang et al., 1997, Chang et al, 2002, Roecker & McGillem, 1998) Therefore, in spite of its limitations, the fusion formula has been widely used and is superior to the centralized filtering in real applications The aforementioned papers have not focused on prediction problem, but most of them have considered only distributed filtering in multisensory continuous and discrete dynamic models Direct generalization of the distributed fusion filtering algorithms to the prediction problem is impossible The distributed prediction requires special algorithms one of which for discrete-time systems was presented in (Song et al 2009) In this paper, we generalize the results of (Song et al 2009) on mixed continuous-discrete systems The continuous-discrete 40 Discrete Time Systems approach allows system to avoid discretization by propagating the estimate and error covariance between observations in continuous time using an integration routine such as Runge-Kutta This approach yields the optimal or suboptimal estimate continuously at all times, including times between the data arrival instants One advantage of the continuousdiscrete filter over the alternative approach using system discretization is that in the former, it is not necessary for the sample times to be equally spaced This means that the cases of irregular and intermittent measurements are easy to handle In the absensce of data the optimal prediction is given by performing only the time update portion of the algorithm Thus, the primary aim of this paper is to propose two distributed fusion predictors using fusion formula with matrix weights, and analysis their statistical properties and relationship between them Then, through a comparison with an optimal centralized predictor, performance of the novel predictors is evaluated This chapter is organized as follows In Section 2, we present the statement of the continuous-discrete prediction problem in a multisensor environment and give its optimal solution In Section 3, we propose two fusion predictors, derived by using the fusion formula and establish the equivalence between them Unbiased property of the fusion predictors is also proved The performance of the proposed predictors is studied on examples in Section Finally, concluding remarks are presented in Section Statement of problem – centralized predictor We consider a linear system described by the stochastic differential equation x t = Ft x t + G t v t , t ≥ , (1) where x t ∈ ℜn is the state, v t ∈ ℜq is a zero-mean Gaussian white noise with covariance T E v t vs = Q t δ ( t-s ) , and Ft ∈ ℜn×n , Gt ∈ ℜn×q , and Qt ∈ ℜq×q Suppose that overall discrete observations Yt k ∈ ℜm at time instants t , t , are composed ( ) of N observation subvectors (local sensors) y(1) , ,y(N) , i.e., t t k k T T Ytk =[y(1) … y(N) ]T , t t k (2) k where y(i) , i=1,… ,N are determined by the equations t k y(1) =H(1)x tk +w(1) , y(1) ∈ ℜm1 , t t t t k k k k               (3) y(N) =H(N)xt k +w(N) , y(N) ∈ ℜmN , t t t t k k k k k=1,2, ; t k+1 >t k ≥ t =0 ; m=m + +m N , { } where y(i) ∈ ℜm i is the local sensor observation, H(i) ∈ ℜn×m i , and w(i) ∈ ℜm i , k = 1, 2, tk t t k are zero-mean white Gaussian sequences, w(i) ~ tk initial state x is Gaussian, x0 ~ mutually uncorrelated ( x0 ,P0 ) , and ( k 0,R (i) tk ) , i=1, ,N The distribution of the { } x , v t , and w(i) , i = 1, ,N are assumed t k 41 Distributed Fusion Prediction for Mixed Continuous-Discrete Linear Systems A problem associated with such systems is to find the distributed weighted fusion predictor ˆ x t+Δ , Δ ≥ of the state x t+Δ based on overall current sensor observations { } Yttk = Yt1 , ,Ytk , t < t k ≥ t = , where i is the index of subsystem Then by the analogy with the centralized prediction ˆ t+Δ equations (5), (6) the optimal local predictor x(i) based on the overall local observations y(i) , ,y(i) , t k ≤ t ≤ t + Δ satisfies the following time update and observation update t1 tk equations: { } ⎧ x(i) =Fs x(i) , t k ≤ s ≤ t+Δ , x(i) =x(i) , ˆs ˆ s=t ˆ t ⎪ ˆs k k ⎨ (ii) (ii) (ii) (ii) (ii) T ⎪ Ps =FsPs +Ps Fs +Q s , Ps=t k =Pt k , ⎩ (8) ˆt where the initial conditions x(i) and its error covariance Pt(ii) are given by the continuousk k discrete Kalman filter equations Time update between observations : ⎧ x(i)- =F x(i)- , t ≤ τ ≤ t , x(i)- =x(i) , ˆ ˆ τ=t ˆt τ τ k-1 k ⎪ ˆτ k-1 k-1 ⎨ (ii) (ii) (ii) (ii) (ii) T ⎪ Pτ =Fτ Pτ +Pτ Fτ +Q τ , Pτ=t =Pt , k-1 k-1 ⎩ (9a) Observation update at time t k : ( ( ) ⎧ x(i) =x(i)- +L(i) y(i) -H(i) x(i)- , ˆ ˆ ˆ tk tk tk tk ⎪ tk tk ⎪ ⎪ (i) (ii)- (i)T (i) (ii)- (i)T (i) ⎨ L tk =Pt k H t k H t k Ptk H tk +R t k ⎪ ⎪ P(ii) = I -L(i) H(i) P(ii)- n tk tk tk ⎪ tk ⎩ ( ) ) -1 , (9b) ˆ t ˆ s=t ˆ t+Δ ˆ s=t+Δ estimates, Thus from (8) we have N local filtering x(i) =x(i) and prediction x(i) = x(i) (ii) and corresponding error covariances Pt(ii) and Pt+Δ for i=1, ,N and t ≥ t k Using these values we propose two fusion prediction algorithms Distributed Fusion Prediction for Mixed Continuous-Discrete Linear Systems 43 3.1 The fusion of local predictors (FLP Algorithm) ˆ t+Δ The fusion predictor xFLP of the state x t+Δ based on the overall sensors (2), (3) is ˆ t+Δ constructed from the local predictors x(i) , i = 1, ,N by using the fusion formula (Zhou et al., 2006, Shin et al., 2006): N ˆ x FLP =∑ a(i) x(i) , t+Δ t+Δ t+Δ i=1 N ∑ a(i) =I n , t+Δ (10) i=1 where a(1) ,… ,a(N) are n × n time-varying matrix weights determined from the meant+Δ t+Δ square criterion, ⎡ N J FLP =E ⎢ xt+Δ - ∑ a(i) x(i) t+Δ t+Δ t+Δ ⎢ i=1 ⎣ 2⎤ ⎥ ⎥ ⎦ (11) ˆ t+Δ The Theorems and completely define the fusion predictor xFLP and its overall error FLP ˆ t+Δ covariance Pt+Δ =cov(xFLP ,xFLP ), x FLP =x t+Δ -xFLP t+Δ t+Δ t+Δ ˆ t+Δ ˆ t+Δ Theorem 1: Let x(1) ,… ,x(N) are the local predictors of an unknown state x t+Δ Then a The weights a(1) ,… ,a(N) satisfy the linear algebraic equations t+Δ t+Δ N N i=1 i=1 (ij) (iN) ∑ a(i) ⎡Pt+Δ -Pt+Δ ⎤ =0, ∑ a(i) =In , t+Δ ⎣ t+Δ ⎦ b j=1,… ,N-1; (12) (ii) ˆ t+Δ The local covariance Pt+Δ =cov(x(i) ,x(i) ) , x(i) =x t+Δ -x(i) satisfies (8) and local crosst+Δ t+Δ t+Δ ij ) ( j) ( ) covariance Pt(+Δ = cov( xt i+Δ , xt +Δ ) , i ≠ j describes the time update and observation update equations: ⎧ P(ij)- =F P(ij)- +P(ij)- F T +Q , P(ij)- =P(ij) , t ≤ τ ≤ t , τ τ τ τ τ k-1 k τ=t k-1 t k-1 ⎪ τ T ⎪ (ij) ⎪ (ij) (j) (j) (i) (i) ⎨ Pt k = I n +L t k H tk Pt k I n +L t k H tk , t=t k , ⎪ (ij) ⎪ Ps(ij) =FsPs(ij) +Ps(ij)FsT +Q s , Ps=t =Pt(ij) , t k ≤ s ≤ t+Δ; k k ⎪ ⎩ ( c ( ) ) (13) FLP The fusion error covariance Pt+Δ is given by N (ij) FLP Pt+Δ = ∑ a(i) Pt+Δ a(j) t+Δ t+Δ T (14) i,j=1 ˆ t+Δ ˆ t+Δ ˆ t+Δ Theorem 2: The local predictors x(1) ,… ,x(N) and fusion predictor xFLP are unbiased, i.e., ( ) ( ) ˆτ ˆ t+Δ E x(i) =E ( x τ ) and E xFLP =E ( x t+Δ ) for ≤ τ ≤ t+Δ The proofs of Theorems and are given in Appendix Thus the local predictors (8) and fusion equations (10)-(14) completely define the FLP algorithm In particular case at N = , formulas (10)-(12) reduce to the Bar-Shalom and Campo formulas (Bar-Shalom & Campo, 1986): 44 Discrete Time Systems ˆ t+Δ t+Δ ˆ t+Δ t+Δ ˆ t+Δ x FLP =a(1) x(1) +a(2) x(2) , -1 (22) (21) (11) (22) (12) (21) a(1) = ⎡Pt+Δ -Pt+Δ ⎤ ⎡Pt+Δ +Pt+Δ -Pt+Δ -Pt+Δ ⎤ , t+Δ ⎣ ⎦⎣ ⎦ (15) (11) (12) (11) (22) (12) (21) -1 a(2) = ⎡Pt+Δ -Pt+Δ ⎤ ⎡Pt+Δ +Pt+Δ -Pt+Δ -Pt+Δ ⎤ t+Δ ⎣ ⎦⎣ ⎦ Further, in parallel with the FLP we offer the other algorithm for fusion prediction 3.2 The prediction of fusion filter (PFF Algorithm) This algorithm consists of two parts The first part fuses the local filtering estimates ˆt ˆt x(1) ,… ,x(N) Using the fusion formula, we obtain the fusion filtering (FF) estimate k k N ˆt ˆ x FF =∑ b(i) x(i) , t t k i=1 k k N ∑ b(i) =I n , t k i=1 (16) where the weights b(1) ,… ,b(N) not depend on lead Δ tk tk ˆt In the second part we predict the fusion filtering estimate xFF using the time update k ˆ t+Δ prediction equations Then the fusion predictor xPFF and its error covariance PFF ˆ t+Δ Pt+Δ =cov(xPFF ,xPFF ), xPFF =x t+Δ -xPFF satisfy the following equations: t+Δ t+Δ t+Δ ⎧ xPFF =Fs xPFF , t k ≤ s ≤ t+Δ , xPFF =xFF , ˆs ˆ s=t ˆ t ⎪ ˆs k k ⎨ PFF PFF PFF T PFF Ps =Fs Ps +Ps Fs +Q s , Ps=t k =PtFF ⎪ k ⎩ (17) Next Theorem completely defines the PFF algorithm ˆt ˆt Theorem 3: Let x(1) ,… ,x(N) are the local filtering estimates of an unknown state x t Then k a The weights k b(1) ,… , b(N) tk tk satisfy the linear algebraic equations N N ⎡ ⎤ ∑ b(i) ⎣Pt(ij) -Pt(iN) ⎦ =0, ∑ b(i) =I n , t t i=1 b c k k k i=1 k j = 1,… ,N − 1; (18) The local covariance Pt(ii) and cross-covariance Pt(ij) in (18) are determined by equations (9) and k k (13), respectively; ˆt The initial conditions xFF and PtFF in (17) are determined by (16) and formula k k N PtFF = ∑ b(i) Pt(ij)b(j) , t t k i,j=1 T k k k (19) respectively; ˆ t+Δ ˆ t+Δ d The fusion predictor xPFF in (17) is unbiased, i.e., E(xPFF )=E(x t+Δ ) The proof of Theorem is given in Appendix 3.3 The relationship between FLP and PFF ˆ t+Δ ˆ t+Δ Here we establish the relationship between the prediction fusion estimates xFLP and xPFF determined by (10) and (16), respectively Distributed Fusion Prediction for Mixed Continuous-Discrete Linear Systems 45 ˆ t+Δ ˆ t+Δ Theorem 4: Let xFLP and xPFF be the fusion prediction estimates determined by (10) and (16), respectively, and the local error covariances Ps(ij) , t k ≤ s ≤ t+Δ , i,j = 1, ,N are nonsingular Then ˆ t+Δ ˆ t+Δ xFLP =xPFF for Δ > (20) The proof of Theorem is given in Appendix (ij) Remark (Uniqueness solution): When the local prediction covariances Pt+Δ , i,j = 1, ,N are nonsingular, the quadratic optimization problem (11) has a unique solution, and the wights a(1) ,… ,a(N) are defined by the expressions (11) The same result is true for the covariance t+Δ t+Δ Pt(ij) and the weights b(1) ,… , b(N) (Zhu et al., 1999, Zhu, 2002) tk tk k Remark (Computational complexity): According to Theorem 4, both the predictors FLP and PFF are equivalent; however, from a computational point of view they are different To predict the state x t+Δ using FLP we need to compute the matrix weights a(1) ,… ,a(N) for t+Δ t+Δ each lead Δ > This contrasts with PFF, wherein the weights b(1) ,… , b(N) are computed tk tk only once, since they not depend on the leads Δ Therefore, FLP is deemed more complex than PFF, especially for large leads Remark (Real-time implementation): We may note that the local filter gains L(i) , the error t (ij) cross-covariances Pt(ij) , Pt+Δ , and the weights a(i) , b(i) may be pre-computed, since they t+Δ tk not depend on the current observations y(i) , i = 1, ,N , but only on the noises statistics Q t tk and R (i) , and system matrices Ft , G t , H(i) , which are part of the system model (1), (3) t t Thus, once the observation schedule has been settled, the real-time implementation of the ˆt fusion predictors FLP and PFF requires only the computation of the local estimates x(i) , ˆ t+Δ ˆ t+Δ ˆ t+Δ x(i) , i = 1, ,N and final fusion predictors xFLP and xPFF ˆt ˆ t+Δ Remark (Parallel implementation): The local estimates x(i) , x(i) , i = 1, ,N are separated for different sensors Therefore, they can be implemented in parallel for various types of observations y(i) , i = 1, ,N t Examples 4.1 The damper harmonic oscillator motion System model of the harmonic oscillator is considered in (Lewis, 1986) We have 1⎤ ⎡ ⎡0 ⎤ * xt = ⎢ ⎥ xt + ⎢ ⎥ vt , ≤ t ≤ t , 1⎦ -ω n -2α ⎥ ⎢ ⎣ ⎣ ⎦ (21) where xt =[x1,t x 2,t ]T , and x1,t is position, x 2,t is velocity, and v t is zero-mean white Gaussian noise with intensity q , E(v t vs )=qδ t-s , x ~ (x0 ,P0 ) Assume that the observation system contains N sensors which are observing the position x1,t Then we have y(1) =H(1)x tk +w(1) , t t t k k k (22)             y(N) =H(N)x t k +w(N) , tk tk tk * 0=t