Reduced-order filter for stochastic bilinear systems with multiplicative noise

Reduced-order filter for stochastic bilinear systems with multiplicative noise S Halabi H Souley Ali H Rafaralahy M Zasadzinski M Darouach Université Henri Poincaré Nancy I CRAN UMR 79 CNRS IU de Longwy 186 rue de Lorraine 544 Cosnes et Romain FRANCE e-mail : {shalabimzasad}@iut-longwyuhp-nancyfr Abstract his paper deals with the design of a reducedorder H filter for a stochastic bilinear systems with a prescribed H norm criterion he problem is transformed into the search of a unique gain matrix by using a Sylvester-like condition on the drift term he considered system is bilinear in control and with multiplicative noise in the state and in the measurement equations he approach is based on the resolution of LMI and is then easily implementable Keywords Reduced-order H filter Itô s formula Stochastic systems Bilinear systems Lyapunov function I Introduction he bilinear systems represent sometimes a good mean to physical systems modeling when the linear representation is not sufficiently significant he stochastic systems get a great importance during the last decades as shown by numerous references (Kozin 1969; Has minskii 198; Florchinger 1995; Mao 1997; Carravetta et al ; Germani et al ; Xu and Chen ) Generally bilinear stochastic system designs a stochastic system with multiplicative noise instead of additive one (Carravetta et al ; Germani et al ) he full and the reduced-order H filtering for stochastic systems with multiplicative noise has been treated in many papers (Hinrichsen and Pritchard 1998; Gershon et al 1; Xu and Chen ; Stoica ) Notice that the measurement equation in (Xu and Chen ; Stoica ) is not corrupted by noise he problem is solved in terms of two LMIs and a coupling non convex rank constraint In this paper the problem of reduced-order H filtering for a larger class of stochastic systems than those studied in the papers cited above is considered since the studied systems are with multiplicative noise and multiplicative control input (the bilinearity is also between the state and the control input) he measurements are subjected to a multiplicative noise too Notice that as in the deterministic case the multiplicative control input affects the observability of the system he purpose is to design a reduced-order H filter for such a system We first use a unbiasedness (decoupling) condition on the drift part of the estimation error and a change of variable on the control input hen applying the Itô formula and LMI method permit to reduce the problem to the search of a unique gain matrix he reduced-order stochastic filter matrices are then computed using this gain hroughout the paper E represents expectation operator with respect to some probability measure P X Y = X Y represents the inner product of the vectors X Y IR n herm(a) stands for A + A L `Ω IR k is the space of square-integrable IR k -valued functions on the probability space (Ω F P) Ω is the sample space F is a σ-algebra of subsets of the sample space called events and P is the probability measure on F (F t) t denote an increasing family of σ-algebras (F t) F We also denote by L b `[ ) ; IR k the space of non-anticipatory square-integrable stochastic process f() = (f(t)) t [ ) in IR k with respect to (F t) t [ ) satisfying f b L = E jz ff f(t) d t < is the well-known Euclidean norm II Problem statement Let us consider the following stochastic bilinear system 8 d x(t) = (A tx(t) + u 1(t)A t1x(t)) d t >< +B d t + A wx(t) d w (t) d y(t) = Cx(t) d t + J >: 1x(t) d w 1(t) z(t) = Lx(t) x(t) IR n is the state vector y(t) IR p is the output u 1(t) IR is the control input z(t) IR r is a linear combination of the state vector with r < n and IR q is the perturbation signal Without loss of generality L is assumed to be a full row rank matrix w i(t) is a Wiener process verifying (Has minskii 198) E (d w i(t)) = E(d w i(t) ) = d t for i = 1 E (d w (t) d w 1(t)) = E(d w 1(t) d w (t)) = ϕ d t with ϕ < 1 (1) (a) (b) As in the most cases for physical processes we assume that the stochastic bilinear system (1) has known bounded control input ie u 1(t) Γ IR Γ = {u 1(t) IR u 1 min u 1(t) u 1 max} () he study made here can be easily generalized for the case there are m control inputs First we introduce the following definition and assumption Definition 1 (Kozin 1969; Has minskii 198) he stochastic system (1) with is said to be asymptotically mean-square stable if all initial states x() yields lim E t x(t) = u 1(t) Γ (4)

Assumption 1 he stochastic bilinear system (1) is assumed to be asymptotically mean-square stable In this paper the aim is to design a reduced-order filter in the following form d η(t) = (M + u 1(t)M 1) η(t) d t + (N + u 1(t)N 1) d y(t) (5) η(t) IR r is the filter state with r < n and the matrices M i and N i (for i = 1) are to be determined hen the following problem is considered Problem 1 Given a real γ > the goal is to design a asymptotically mean-square stable reduced-order H filter (5) such that the augmented state [x (t) e (t)] is asymptotically mean-square stable and the following H performance e(t) L b γ L b (6) is achieved from the disturbance to the filtering error e(t) = z(t) η(t) Let us consider the following estimation error e(t) = Lx(t) η(t) (7) So the estimation error dynamics becomes d e(t) = (M + M 1u 1(t)) e(t) d t + LB d t + {(LA t M L N C) + (LA t1 M 1L N 1C)u 1(t)} x(t) d t + LA wx(t) d w (t) ((N + u 1(t)N 1) J 1x(t) d w 1(t) (8) In order to supress the direct effect of the state x(t) on the drift part of the filtering error we consider the following Sylvester-like conditions LA ti M il N ic = i = 1 (9) Let us consider the following augmented state vector ξ (t) = ˆx (t) e (t) (1) hen under (9) the dynamics of the augmented system is given by with d ξ(t) = (A t + A t1u 1(t)) ξ(t) d t + B d t + A wξ(t) d w (t) + (A w1 + A wu 1(t)) ξ(t) d w 1 (11) Ati A ti = for i = 1 M i B Aw B = A w = LB LA w A w1 = A w = N J 1 N 1J 1 In the sequel the relations (9) are used to express the filter matrices through a single gain matrix In fact since L is a full row rank matrix relations (9) are equivalent to (LA ti M il N ic) ˆL (I n L L) = for i = 1 (1) L is a generalized inverse of matrix L satisfying L = LL L (Lancaster and ismenetsky 1985) (since rank L = r we have LL = I r) Relations (1) give = LA til M i N icl for i = 1 (14a) = LA i N ic for i = 1 (14b) A i = A ti(i n L L) for i = 1 (15a) C = C(I n L L) he relation (14a) gives (15b) M i = A i N ic for i = 1 (16) A i = LA til for i = 1 (17a) C = CL he relation (14b) becomes (17b) KΣ = LA (18) K = ˆN N 1 (19) A = ˆA A 1 () C Σ = C (1) and a general solution to equation (18) if it exists is given by K = LA Σ + Z(I p Σ Σ ) () Z = ˆZ Z 1 () is an arbitrary matrix of appropriate dimensions III ranformation of the bilinear system filtering problem into an uncertain one As in (Zasadzinski et al ) let us introduce a change of variable on the control u 1(t) as follows α 1 IR and σ 1 IR are given by u 1(t) = α 1 + σ 1ε 1(t) (4) α 1 = 1 (u1min + u1max) σ1 = 1 (u1max u1min) (5) he new uncertain variable is ε 1(t) Γ IR the polytope Γ is defined by Γ = {ε 1(t) IR ε 1min = 1 ε 1(t) ε 1max = 1} (6)

hen the error dynamics (8) can be rewritten as and d e(t) = A t ZC t + ( e A t Z e C t) ε(ε 1(t))H e e(t) d t + B d t + A wx(t) d w (t) + `A w(11) ZA w(1) +( e A w(11) Z e A w(1) ) x(ε 1(t))H x x(t) d w 1(t) (7) A t = A + α 1A 1 LA Σ Λ ea t = σ 1A 1 LA Σ Λ C t = (I p Σ Σ )Λ e Ct = (I p Σ Σ )Λ B 1 = LB A w = LA w A w(11) = LA Σ Ψ α A w(1) = (I p Σ Σ )Ψ α ea w(11) = LA Σ Ψ σ e Aw(1) = (I p Σ Σ )Ψ σ CL J1 Λ = α 1CL Ψ α = Ψ σ = αj 1 σj 1 H e = I r H x = I n ε(ε 1(t)) = ε 1(t)I r x(ε 1(t)) = ε 1(t)I n Using the definition (6) the matrix ε(ε 1(t)) and x(ε 1(t)) satisfy ε1 (ε 1(t)) 1 and x(ε 1(t)) 1 (8) Using (4) the system state equation (see (1)) becomes d x(t) = (A t + α 1A t1 + σ 1ε 1(t)A t1) x(t) d t + B d t + A wx(t) d w (t) (9) So the augmented system (11) is rewritten as d ξ(t) = bat + A b t(t) ξ(t) d t + B b d t + A b wξ(t) d w (t) + baw1 + A b w1(t) ξ(t) d w 1(t) () At + α 1A t1 ba t = A t ZC t A b t(t) = H 1 ξ (ε 1(t))H t B bb = Aw b = B 1» ba w1 = A w(11) ZA w(1) Aw A w A b w1(t) = H ξ (ε 1(t))H w " # " # σ 1A t1 H 1 = At e ZC e H = t ea w(11) ZA e w(1) In H t = H w = ˆI n ξ (ε 1(t)) = ε 1(t) I r Notice that from (6) ξ (ε 1(t)) satisfy ξ (ε 1(t)) 1 (1) IV Synthesis of the reduced-order filter Consider the following system obtained from () 8 >< d ξ(t) = bat+ A t(t) b ξ(t) d t+ B b d t + A >: b w ξ(t)d w (t)+ baw1+ A b w1(t) ξ(t) d w 1(t) e(t) = Cξ(t) b () b C = ˆ I r hen the following theorem is given for the filter synthesis heorem 1 he reduced-order H filtering problem 1 is solved for the system (1) with the filter (5) such that the augmented system () is asymptotically mean-square stable and verifies the H performance (6) if for some reals µ 1 > µ > and µ > there exist matrices P 1 = P 1 > IR n n P = P > IR r r P IR n r G IR r p and G IR n p such that 6 4 (11) (1) P 1 B +P B 1 σ 1 P 1 A t1 (1) () P B +P B 1 σ 1 P A t1 B P 1+B 1 P B P +B 1 P γ I q σ 1 A t1 P 1 σ 1 A t1 P µ 1 I n ea t P e C t G A e t P C e t G (16) (17) (18) (19) (11) P At e G Ct e (16) (17) (18) (19) (11) P At e G Ct e µ 1 I r P 1 P P P µ I n P 1 P (119) P P (111) (119) (111) µ I n < () 7 5 (1 1) = (µ 1 + µ + ϕµ )I n + herm {P 1A α1 + ϕ A w `PA w(11) G A w(1) + A w `PA w(11) G A w(1) (1 ) = A α1p + P A t G C t ( ) = herm (P A t G C t) + (1 + µ 1) I r (1 6) = A wp 1 + A wp (1 7) = A wp + A wp (1 8) = ϕ 1 A w P Aw(11) e G Aw(1) e P Aw(11) e G Aw(1) e +A w (1 9) = A w(11)p A w(1)g (1 1) = A w(11)p A w(1)g (11 9) = P e Aw(11) G e Aw(1) (11 1) = P e Aw(11) G e Aw(1) A α1 = A t + α 1A t1 and such that the gain matrices G and G are the solution of the following equation» G G =» P P Z (4)

Proof Consider the following Lyapunov function V (ξ) = ξ Pξ (5) P1 P P = P (6) P Applying Itô formula (Mao 1997) to the system () (or ()) we get d V (ξ(t)) = LV (ξ(t)) d t + ξ (t)pψ(t)ξ(t) (7) Ψ(t) = A b w d w (t) + baw1 + A b w1(t) d w 1(t) (8) and LV (ξ(t)) d t=ξ (t) bat+ A b t(t) ξ(t)+ B b d t + ξ (t) PΨ(t) Ψ(t) ξ(t) (9) By replacing (8) and (9) the relation (7) becomes d V (ξ(t)) = ξ (t)p bat + A b t(t) ξ(t) + B b d t + ξ (t) b A wp b A wξ(t) d w (t) + ξ (t) baw1 + A w1(t) b P baw1 + A b w1(t) ξ(t) d w 1(t) + ξ (t) A b wp baw1 + A b w1(t) ξ(t) d w (t) d w 1(t) + ξ (t) baw1 + b A w1(t) P b Awξ(t) d w 1(t) d w (t) + ξ (t)pa b wξ(t) d w (t) + ξ (t)p baw1 + A b w1(t) ξ(t) d w 1(t) (4) " PA Θ 1 = b t + A b tp + C C PB b # bb P γ I q H + µ t H t 1 + µ 1 PH 1 H 1P 1 " # ba + w1`p 1 µ 1 1 HH b Aw1 H + µ w H w " H + ϕµ w H w ba + w P b # A w " ϕµ 1 ba + w PH H PA b # w " ba + ϕ w PA b w1 + A b w1pa b # w (45) Now applying the Schur lemma (Boyd et al 1994) Θ 1 can be rewritten as (1 1) PB b PH 1 A b w P bb P γ I q H1 P µ 1I n+r PA b w P ϕ 1 H PA 6 b w 4 PA b w1 ϕ 1 A b w PH A b w1 P µ I n+r 7 P PH 5 H P µ I n (46) (1 1) = PA b t + A b tp + C b C b + (µ + ϕµ )Hw H w + µ 1Ht H t + ϕ ba w PA b w1 + A b w1pa b w (47) Using the majoration lemma (Wang et al 199) it can be Once the LMI () is verified the asymptotic mean-square shown that stability of the system () for can be proved using Schur lemma and the same method of (Souley Ali et ξ (t)p A b al 5) t(t)ξ(t) Now consider the following performance index ξ (t) µ 1 1 PH 1H1 P + µ 1Ht H t ξ(t) (41) Z baw1+ A b P w1(t) baw1+ A w1(t) b J ξv = E ξ (t) b C Cξ(t) b γ v (t) d t (48) A b 1 w1 P 1 µ 1 H H baw1+µ Writting Hw J ξv as H w (4) ξ (t) A b wp A b w1(t)ξ(t) ξ (t) µ 1 ba wph H PA b w + µ Hw H w ξ(t) (4) J ξv = Z n E ξ (t) C b Cξ(t) b γ v (t) d t + d V (ξ(t)))} E (V (ξ(t)) t= + E (V (ξ(t)) t= (49) Now taking the expectation of (4) (see (Mao 1997)) and using the relations () and the last three inequalities then E{d V (ξ(t))} can be bounded as E {d V (ξ(t))} E jˆξ(t) Θ ff ξ(t) 1 d t (44) Or since E (V (ξ(t)) t= = because ξ() = and E (V (ξ(t)) t= this implies J ξv Z n E ξ (t) C b Cξ(t) b γ v (t) d t + d V (ξ(t)))} (5)

Now if the LMI () holds then applying Schur lemma yields " Θ PB b # " # bc C b bb + P γ < (51) I q {z } Π with he following figures show the simulation results of the augmented system () he state x(t) and the estimation error e(t) are plotted he disturbance signal is presented with the error plots he simulation is made for the control u 1(t) = 5 sin(t) + and the covariance factor between the Wiener processes defined in (b) ϕ = 15 Θ = P b A t + b A tp + µ 1H t H t + (µ + ϕµ )H w H w herefore J ξv + ϕ( A b wpa b w1 + A b w1pa b w) + µ 1 1 PH1 H 1P+ Z ba wp b A w + ϕµ 1 E ˆξ(t) Π ξ(t) d t + ˆξ(t) " b C b C γ I q ba w PH H P b A w #»ξ(t)! d t < so if the LMI () holds the asymptotic mean-square stability and the H performance are proved V Numerical example Consider the stochastic bilinear system (1) and suppose that the matrices have the following numerical value 15 1 1 1 A t = 4 5 5 1 5 B = 4 1 5 6 5 6 5 1 1 A t1 = 4 5 5 15 1 A w = 4 5 15 1 1 1 C = L = 1 1 1 J 1 = 1 he control u 1(t) is defined as in () with u 1 min = 5 u 1(t) u 1 max = 6 and the initial state ξ() = [ x () e () ] ξ() = ˆ 1 5 1 5 he gain Z is obtaind for γ = and µ 1 = 7468 µ = 57 and µ = 476 and is then given by Z = 6179565 6191468 478 4719 18174 1819847 76 796 Finally the matrices of the reduced-order filter (5) are 991 4791 41 M = M 1 = 486 76 114 146 491 791 171 9 N = N 1 = 586 6 is Fig Fig 1 ime [sec] he actual state x(t) ime [sec] he error e(t) and the disturbance VI Conclusion his paper provided a solution to the reduced-order H filtering problem for bilinear stochastic systems with multiplicative noise he approach is based on a change of variable on the control input and on the using of a Sylvesterlike condition on the drift term to transform the problem into a robust reduced-order stochastic filtering one Using the LMI method and the Itô formula we reduced the problem to the search of a unique gain matrix hen the filter matrices are computed through this gain References Boyd SP L El Ghaoui E Féron and V Balakrishnan (1994) Linear Matrix Inequality in Systems and Control heory SIAM Philadelphia Carravetta F A Germani and MK Shuakayev () A new suboptimal approach to the filtering problem for bilinear stochastic differential systems SIAM J Contr Opt 8 1171 1 Florchinger P (1995) Lyapunov-like techniques for stochastic stability SIAM J Contr Opt 1151 1169 Germani A C Manes and P Palumbo () Linear filtering for bilinear stochastic differential systems with unknown inputs IEEE rans Aut Contr 47 176 17

Gershon E DJN Limebeer U Shaked and I Yaesh (1) Robust H filtering of stationary continuous-time linear systems with stochastic uncertainties IEEE rans Aut Contr 46 1788 179 Has minskii RZ (198) Stochastic Stability of Differential Equations Siijthoff and Noordhoff Aplhen aan den Rijn he Netherlands Hinrichsen D and AJ Pritchard (1998) Stochastic H SIAM J Contr Opt 6 154 158 Kozin F (1969) A survey of stability of stochastic systems Automatica 5 95 11 Lancaster P and M ismenetsky (1985) he heory of Matrices nd ed Academic Press Orlando USA Mao X (1997) Stochastic Differential Equations & Applications Horwood London Souley Ali H H Rafaralahy M Zasadzinski S Halabi and M Darouach (5) Observer design for a class of stochastic bilinear systems with multiplicative noise In: Proc IEEE American Contr Conf Portland USA Stoica Adrian () H filtering of signals subjected to multiplicative white noise In: Proc riennal IFAC World Congress Barcelona Spain Wang Y L Xie and CE De Souza (199) Robust control of a class of uncertain nonlinear systems Syst & Contr Letters 19 19 149 Xu S and Chen () Reduced-order H filtering for stochastic systems IEEE rans Sign Proc 5 998 7 Xu S and Chen () Robust H filtering for uncertain impulsive stochastic systems under sampled measurements Automatica 9 5 516 Zasadzinski M E Magarotto H Rafaralahy and H Souley Ali () Residual generator design for singular bilinear systems subjected to unmeasurable disturbances : an LMI approach Automatica 9 7 71