FILTERING OF NONLINEAR STOCHASTIC FEEDBACK SYSTEMS

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1 FILTERING OF NONLINEAR STOCHASTIC FEEDBACK SYSTEMS F. CARRAVETTA 1, A. GERMANI 1,2, R. LIPTSER 3, AND C. MANES 1,2 Abtract. Thi paper concern the filtering problem for a cla of tochatic nonlinear ytem where the drift term may depend either on ome external function open-loop ytem or on the ytem output cloed-loop ytem, through a controller. Such ytem are denoted feedback ytem. The following reult i proven: for feedback ytem, the optimal filter in the open-loop cae remain optimal when the feedback i cloed. 1. Introduction Conider the cla of nonlinear tochatic ytem decribed by the equation: dx t = ft, X t, ut, [,t] dt + bt, X t dw t, dy t = ht, X t dt + BtdW t, 1.1 where X t R n i the ytem tate, Y t R m i the obervation proce, ut, [,t] R p i the input function, generated by ome driving function. f, h are vector function of uitable dimenion. W t R n and W t R m are independent Wiener procee without lo of generality we conider quare diffuion matrice b and B. If in ytem 1.1 the driving function i replaced by the ytem output Y, we obtain the following ytem dx t = ft, X t, ut, Y [,t] dt + bt, X t dw t, dy t = ht, X t dt + BtdW t. 1.2 So, the term ut, Y [,t] repreent a caual map of the obervation proce into the input, decribing a behavior of ome feedback control device the controller. We will refer to ytem 1.1 a the open loop ytem, and to ytem 1.2 a the cloed loop ytem. Let F : R n R n be a function of the ytem tate that define a ignal to be etimated for the open and cloed loop ytem: S t = F X t, 1.3 S t = F X t. 1.4 Aume for every fixed t there i a function Ψ t y [,t] ; [,t], y t, t, t, are continuou vector function valued in R p uch that Ψ t Y [,t] ; [,t] = E S t /Y [,t] Mathematic Subject Claification. 93E3, 93E11. Key word and phrae. Nonlinear filtering, cloed-loop ytem, Giranov theorem, convergence of probability meaure. 1

2 2 F. CARRAVETTA 1, A. GERMANI 1,2, R. LIPTSER 3, AND C. MANES 1,2 Thi i the open-loop filter, i.e. the optimal filter for the open-loop ytem 1.1, forced by the ytem output and by the forcing term. For every t aume alo there exit a function Φ t y [,t] uch that Φ t Y [,t] = E S t /Y [,t], P -a Thi i the cloed-loop filter, i.e. the optimal filter for the cloed-loop ytem 1.2, that i forced by the ytem output only. The following quetion arie: Ψ t Y [,t] ; Y [,t]? = Φ t Y [,t], P -a.., 1.7 tated in other word: if we apply the open-loop filter to the cloed-loop ytem, then doe the etimate agree with the optimal tate-etimate for the cloed-loop ytem? The quetion if 1.7 hold or not i not only intereting by itelf, but i important in many application. For intance, in all cae in which a finite-dimenional filter exit for the open loop ytem ee [3], identity 1.7 prove that the filter remain optimal and finite-dimenional alo when the feedback i cloed. Another intereting application i when Φ t Y [,t] i computed by the Monte-Carlo method via Ψ t Y [,t] ; [,t]. Up to now, the correctne of 1.7 ha been proved only for particular cae of the problem, uch a in the cae of linear-gauian ytem under nonlinear feedback [5] of the type dx t = At, Y [,t] X t dt + F t, Y [,t] dw t, dy t = Ct, Y [,t] X t dt + Gt, Y [,t] dw t, important from an application point of view. In thi paper, we give affirmative anwer to the quetion 1.7 for the nonlinear model 1.1, 1.2, under ome not really retrictive aumption. The paper i organized a follow: ection 2 report the rigorou tatement of the problem i given and ection 3 preent the main theorem. Concluion follow. 2. Problem Statement On a probability pace {Ω, F, P }, conider two independent Wiener procee W t and W t, t [,, of dimenion n and m, repectively, and a random vector X R n. Let F t be the nondecreaing family of σ-algebra generated by {X, W, W, t}. Throughout the paper C [, R q hall denote the pace of R q -valued continuou function over the interval [,. On thi pace, let B q t, t, be the σ-algebra generated by cylinder et of the form { ϕ C[, R q : t k B k ; t k < t; k =1,..., k; k N; B k BR q }, 2.1 where BR q i the Borel σ-algebra of R q. Moreover, let B q = B q t. Let R + be the Borel σ-algebra on R +. Given a proce ξ t, let σ t ξ be the σ-algebra generated by {ξ, t}. t

3 FILTERING OF NONLINEAR STOCHASTIC FEEDBACK SYSTEMS 3 For a given C [, R m, conider the open loop model: dx t = ft, X t, ut, dt + bt, X t dw t, X = X, dy t = ht, X t dt + BtdW t, Y =, S t = F 2.2 X t. Conider alo the cloed loop model: dx t = ft, X t, ut, Y dt + bt, X t dw t, X = X, dy t = ht, X t dt + BtdW t, Y =, S t = F X t. 2.3 In both model the tate pace i R n, the obervation pace i R m and the ignal pace i R n. The independent Wiener procee W t and W t are n and m dimenional, repectively. For model 2.2 and 2.3 we make the following aumption: i the function u : R + C [, ] R m R p i R + B -meaurable m and {Bt m } t - adapted. ii for any t R + the function ft,,, ht,,, F have bounded component; iii there exit an increaing function Lt and a meaure µdt on R +, with µd <, t >, o that here i the Euclidean norm, ft, x, ut, y ft, x, ut, y Lt x x + y y µd, x x, ; ht, x ht, x Lt bt, x bt, x Lt x x 2.4 iv matrice D t := BB t and d t := bb t, x i the tranpoition ymbol are uniformly noningular repectively in R + and in R + R n, with bounded invere; v Open loop filter. There exit a function Ψ : R + C [, ] R m C [, ] R m R n, R + B m B -meaurable m and {Bt m Bt m } t -adapted, uch that Ψ t Y ; = E S t /σ t Y, P -a.., t R vi Cloed loop filter. There exit a function Φ : R + C [, ] R m R n, R + B m -meaurable and {B m t } t -adapted, uch that Φ t Y = E S t /σ t Y, P -a.., t R Note that thank to the aumption of {B t } t -meaurability of the function u, the term ut, Y perform a caual mapping of the obervation proce into the input. Moreover, note that condition iii guarantee exitence and uniquene of trong olution of 2.2 and of 2.3, adapted to F t.

4 4 F. CARRAVETTA 1, A. GERMANI 1,2, R. LIPTSER 3, AND C. MANES 1,2 3. Main Reult The main reult of thi paper i given by the following theorem, that anwer to the quetion 1.7. Theorem 3.1. Conider the open-loop and the cloed-loop nonlinear tochatic model 2.2 and 2.3. Let the aumption i-vi be atified. Then the function Ψ t and Φ t defined in 2.5 and 2.6 are uch that Ψ t Y ; Y = Φ t Y, P -a.., t R Before to give the proof of thi theorem we need to tate ome preliminary reult. Throughout the paper we will ue the following notation ht, x 2 = h t, x BB 1 tht, x. 3.2 t Moreover, for a given proce ξ taking value on C [, R q, we hall denote with µ t ξ the meaure induced by the proce on { } C [, R q, B q t. Let F t : C [, ] R n R n be the bounded function defined by the equality F t z = F zt, where F i the function defining the ignal for ytem 2.2, 2.3. Lemma 3.1. Kallianpur-Striebel formula for Ψ t Y ; For any t the open-loop filter can be written a F Ψ t Y C [, R ; = n tzλ t z, Y µ t dz X Λ, 3.3 C [, R n tz, Y µ t dz X where Λ t X, Y = exp Proof. Conider the proce h, X dy 1 2 h, X 2 d. 3.4 dζ t = BtdW t, ζ =. 3.5 By Theorem 7.2 and comment from Subection after thi theorem in [4], for any t the ditribution of procee X, Y t, X, ζ t are equivalent. Moreover, it i Λ t z, y = dµt X,Y z, y dµ t X,ζ z, y C [, R n C [, R m From thi, the following equation i obtained Λ t z, yµ t X dz = dµt Y y C [, R n dµ t ζ y C [, R m From Theorem 7.23 in [4], and it multi-dimenional analog Lemma 2.3 in [1], it i Ψ t Y ; = F t zρ t z, Y µ t X dz, 3.8 C [, R n

5 FILTERING OF NONLINEAR STOCHASTIC FEEDBACK SYSTEMS 5 with ρ t z, y = dµt / X,Y dµ t Y z, y y. 3.9 dµ t dµ t X,ζ ζ From the expreion of the Radon-Nikodym derivative it follow ρ t z, y = and from thi equation 3.3 follow. Λ t z, y C [, R n Λ tz, yµ t X dz z, y C [, R n C [, R m, 3.1 From aumption i iii, there exit a Q : R + R n C [, R n C [, R m R n, meaurable and { } BR n Bt n Bt m -adapted, uch that the cloed-loop tate proce can be written a: X t = Q t X, W, Y QX, W, Y will denote the proce {Q X, W, Y, R + }. Lemma 3.2. Kallianpur-Striebel formula for Φ t Y. For any t the cloed-loop filter can be written a F R Φ t Y = n C [,t] Qx, w, Y A R n t x, w, Y µ X dxµ t W dw A, 3.12 R n C [,t] R n tx, w, Y µ X dxµ t W dw where { A t x, w, Y = exp h, Q x, w, Y dy 1 2 h, Q x, w, Y } 2 d Proof. A in the proof of Lemma 3.1, apply Theorem 7.2 of [4] to the procee X, W, Y and X, W, ζ. One ha that for all t the ditribution µ t X,W,Y and µ t X,W,ζ are equivalent, and the Radon-Nikodym derivative i dµ t X,W,Y dµ t X,W,ζ x, w, y = A t x, w, y x, w, y R n C [, R n C [, R m where A t i defined in The following equation can be verified A t x, w, yµ X dxµ t W dw = dµt Y y, C [, R n dµ t ζ y C [, R m Uing Theorem 7.23 in [4], and it multi-dimenional analog Lemma 2.3 in [1], it i Φ t Y = F Q x, w, Y γ x, w, Y µ X dxµ t W dw, 3.16 R n C [, R n

6 6 F. CARRAVETTA 1, A. GERMANI 1,2, R. LIPTSER 3, AND C. MANES 1,2 with From thee one ha γ t x, w, y = dµt / X,W,Y dµ t x, w, y Y y dµ t X,,W,ζ dµ t ζ γ t x, w, y = Equation 3.12 follow. A t x, w, y R n C [, R n A tx, w, yµ X dxµ t W dw x, w, y R n C [, R n C [, R m. Let u define the proce Υ a follow { Υ t X, Y = exp h, X dy 1 2 Note that, from 3.11 and 3.13 it i 3.18 h, X } 2 d Υ t QX, W, Y, Y = A t X, W, Y. 3.2 In the following we need to rewrite the expreion of the open and cloed loop filter, given by 3.3 and 3.12, repectively, in a more convenient form related to the underlying probability pace. To thi purpoe, we introduce a copy Ω, F, P of the original probability pace, o that all procee defined on it are independent copie of the original one. We introduce alo random variable and procee on the product probability pace Ω Ω, F F, P P. Let Zω, ω be a random variable defined on the product pace. Let u define the operator Ẽ a follow ẼZω = Zω, ωp d ω Ω For a given proce ξ defined on the original pace, we hall denote by ξ a proce defined on the product pace a ξω, ω = ξ ω. Whenever it doe not caue confuion, we hall ue the ame ymbol ξ to denote both the original proce and it extenion to the product pace: ξω, ω = ξω. On the product pace it i poible to define the proce X Y a follow X Y t = Q t X, W, Y 3.22 With thee poition, recalling alo the definition of Υ given in 3.19, we can rewrite the expreion 3.3 and 3.12 a follow: Ψ t Y ; = Ẽ{ F t X Λ t X, Y } Ẽ { Λ t X }, P -a , Y Φ t Y = Ẽ{ F t X Y Υ t X Y, Y } Ẽ { Υ t X }, P -a Y, Y Now we are in a poition to give the proof of Theorem 3.1.

7 FILTERING OF NONLINEAR STOCHASTIC FEEDBACK SYSTEMS 7 Proof of Theorem 3.1. From expreion 3.23 and 3.24, Theorem 3.1 i proved a oon a it i hown that Λ t X, Y =Y = Υ t X Y, Y, P P -a From definition 3.4 and 3.19 we have Λ t X, Y = exp h, X dy 1 2 Υ t X Y, Y = exp h, X Y dy 1 2 h, X 2 d, 3.26 h, X Y 2 d Let u conider ome σ t X, Y -meaurable function H t and L t uch that H t X, Y = L t X, Y = h, X dy, and σ t X Y, Y -meaurable function H t and L t uch that H t X Y, Y = L t X Y, Y = We can ue Lemma 4.1 of [4] to prove that h, X d, D 1 h, X Y dy, h, XY d. H t X Y, Y = H t X Y, Y, L t X Y, Y = L t X Y, Y. 3.3 by howing that the meaure µ t X and,y µt XY are equivalent.,y A a matter of fact, Theorem 7.19 of [4] guarantee the equivalence of the meaure induced by the procee X, X, Y and X Y, X, Y, that are defined on Ω Ω, F F, P P a follow d X t = ft, X t, ut, dt + bt, X t d W t, X = X, dx t = ft, X t, ut, dt + bt, X t dw t, X = X, dy t = ht, X t dt + BtdW t, Y =, 3.31 d X Y t = ft, X Y t, ut, Y dt + bt, X Y t d W t, XY = X, dx t = ft, X t, ut, Y dt + bt, X t dw t, X = X, dy t = ht, X t dt + BtdW t, Y = Since µ t X and,y µt XY are marginal ditribution of,y µt X and,x,y µt XY, repectively, their equivalence follow a well.,x,y

8 8 F. CARRAVETTA 1, A. GERMANI 1,2, R. LIPTSER 3, AND C. MANES 1,2 4. Concluion The contribution of thi paper i Theorem 3.1, that repreent a general property of tochatic ytem that can be informally expreed in thee word: whenever the optimal filter i available for a given ytem in open-loop, the ame filter will work optimally on the ame ytem in cloed-loop. Among the implication of Theorem 3.1 there i the reult that any ytem that admit a finite dimenional filter in open-loop, admit a finite dimenional filter alo in cloed-loop. Acknowledgment : The author gratefully acknowledge the careful review of an anonymou referee, who alo pointed out a mitake in the original verion. Due to hi comment the paper ha been ignificantly improved. Reference [1] B.D.O. Anderon, J.B. Moore, Optimal Filtering, Prentice Hall, Inc., Englewood Cliff, N.J., [2] A.V. Balakrihnan, Kalman Filtering Theory, Optimization Software,Inc., Publication Diviion, New York, [3] M. Cohen De Lara, Finite-Dimenional Filter. Part I: The Wei Norman Technique, Finite-Dimenional Filter. Part II: Invariance Group Technique, SIAM J. Contr. & Opt., Vol. 35, No. 3, pp [4] R.S. Lipter, A.N. Shiryayev. Statitic of Random Procee, Vol 1, Springer Verlag, New York, [5] R.S. Lipter, A.N. Shiryayev. Statitic of Random Procee, Vol 2, Springer Verlag, New York, [6] Kallianpur, G., Striebel, C., Etimation of tochatic ytem: Arbitrary ytem proce with additive noie obervation error, Ann. Math. Statit., , pp [7] R.E. Kalman, A New Approach to Linear Filtering and Prediction Problem, J. Baic. Eng., vol. 1, pp , 196. [8] W.J. Kolodziej, R.R. Mohler, State Etimation and Control of Conditionally Linear Sytem, SIAM Journal on Control and Optimization, Vol. 24, No. 3, pp , May [9] W.J. Kolodziej, R.R. Mohler, Conditionally Linear and Non-Gauian Procee, in Non- Gauian Signal Proceing, E. Wegman and S. Schwartz, ed., North-Holland, Amterdam, [1] D. Michel, Régularité de loi conditionelle en théorie du filtrage non-linéaire et calcul de variation tochatique, Journal of functional analyi 41, 1981, pp Itituto di Analii dei Sitemi e Informatica del CNR Viale Manzoni 3, 185 Roma, Italy. fax: , phone: carravetta@iai.rm.cnr.it 2 Dipartimento di Ingegneria Elettrica, Univerità dell Aquila 671 Monteluco di Roio, L Aquila, Italy. fax: , phone: , {germani,mane}@ing.univaq.it 3 Dept. Electrical Engineering-Sytem, Tel Aviv Univerity Tel Aviv, Irael. lipter@eng.tau.ac.il

Problem 1. Construct a filtered probability space on which a Brownian motion W and an adapted process X are defined and such that

Problem 1. Construct a filtered probability space on which a Brownian motion W and an adapted process X are defined and such that Stochatic Calculu Example heet 4 - Lent 5 Michael Tehranchi Problem. Contruct a filtered probability pace on which a Brownian motion W and an adapted proce X are defined and uch that dx t = X t t dt +