SUB-OPTIMAL RISK-SENSITIVE FILTERING FOR THIRD DEGREE POLYNOMIAL STOCHASTIC SYSTEMS

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1 ICIC Express Letters ICIC International c 2008 ISSN X Volume 2, Number 4, December 2008 pp SUB-OPTIMAL RISK-SENSITIVE FILTERING FOR THIRD DEGREE POLYNOMIAL STOCHASTIC SYSTEMS Ma. Aracelia Alcorta-G., Michael Basin Juan J. Maldonado O., Sonia G. Anguiano R. Department of Physical and Mathematical Sciences Autonomous University of Nuevo Leon Cd. Universitaria, San Nicolas de los Garza, Nuevo Leon, 66450, Mexico { aalcorta; mbasin }@fcfm.uanl.mx { matematico one; srostro Received June 2008; accepted September 2008 Abstract. The risk-sensitive filtering problem with respect to the exponential meansquare criterion is considered for stochastic Gaussian systems with third degree polynomial drift terms and intensity parameters multiplying diffusion terms in the state and observations equations. The closed-form suboptimal filtering algorithm is obtained linearizing a nonlinear third degree polynomial system at the operating point and reducing the original problem to the optimal filter design for a first degree polynomial system. The reduced filtering problem is solved using quadratic value functions as solutions to the corresponding Hamilton-Jacobi-Bellman equation. The performance of the obtained risk-sensitive filter for stochastic third degree polynomial systems is verified in a numerical example against the mean-square optimal third degree polynomial filter and extended Kalman-Bucy filter, through comparing the exponential mean-square criteria values. The simulation results reveal strong advantages in favor of the designed risk-sensitive algorithm for large values of the intensity parameters. Keywords: Risk-sensitive filtering, Stochastic systems 1. Introduction. The optimal mean-square filtering theory was initiated by Kalman and Bucy for linear stochastic systems, and then continued for nonlinear systems in a variety of papers (see for example [1-7]). More than thirty years ago, Mortensen [8] introduced adeterministicfilter model which provides an alternative to stochastic filtering theory. In this model, errors in the state dynamics and the observations are modeled as deterministic disturbance functions, and a mean-square disturbance error criterion is to be minimized. Special conditions are given for the existence, continuity and boundlessness of a drift f(x) in the state equation and a linear function h(x) intheobservationone.a concept of the stochastic risk-sensitive estimator, introduced more recently by McEneaney [9], in regard to a dynamic system including nonlinear drift f(x), linear observations, and intensity parameters multiplying diffusion terms in both, state and observation, equations. Again, the exponential mean-square (EMS) criterion, introduced in [10] for deterministic systems and in [11] for stochastic ones, is used instead of the conventional mean-square criterion to provide a robust estimate, which is less sensitive to parameter variations in noise intensity. This paper presents a solution to the risk-sensitive filtering problem with respect to the exponential mean-square criterion for stochastic third degree polynomial systems including intensity parameters multiplying diffusion terms in both, state and observation, equations. The closed-form suboptimal filtering algorithm is obtained linearizing a nonlinear third degree polynomial system at the operating point and reducing the original problem to the optimal filter design for a first degree polynomial (affine) system. The reduced filtering problem is solved seeking quadratic value functions as solutions to the corresponding Hamilton-Jacobi-Bellman equation. Undefined parameters 371

2 372 M. A. ALCORTA, M. BASIN, J. J. MALDONADO AND S. G. ANGUIANO in the value functions are calculated through ordinary differential equations composed by collecting terms corresponding to each power of the state-dependent polynomial in the HJB equation. The closed-form risk-sensitive filter equations are explicitly obtained. The performance of the obtained risk-sensitive filter for stochastic third degree polynomial systems is verified in a numerical example against the mean-square optimal third degree polynomial filter and extended Kalman-Bucy filter, through comparing the exponential mean-square criteria values. The simulation results reveal strong advantages in favor of the designed risk-sensitive algorithm for large values of the intensity parameters multiplying diffusion terms in state and observation equations. Tables of the criteria values and simulation graphs are included. 2. Problem Statement and Preliminaries Optimal risk-sensitive filtering problem. Consider the following stochastic diffusion model, for the state process X t : dx t = f(x t )dt + dw t, (1) where f(x t ) represents the nominal dynamics. The observation process Y t satisfies the equation: dy t = h(x t )dt + d W t. Here, is a parameter and W and W are independent Brownian motions, which are also independent of the initial condition X 0.X 0 has probability density k exp( 1 φ(x 0 )) with a certain constant k. The exponential mean-square cost function to be minimized over possible estimates m t is given by: Z T J = loge{exp 1 (x t m t ) T (x t m t )dt/y t }, (2) 0 where E(ξ/Y t ) is the conditional expectation of a random variable ξ with respect to the observations process Y t. In the rest of the paper, the assumptions (A1)-(A4) from [12] hold. Let q(t,x) denote the unnormalized conditional density of X T, given observations Y t for 0 t T. It satisfies the Zakai stochastic PDE, in a sense made precise in [13], Sec. 7. Since the normalizing constant k above is unimportant for q, it is assumed that q(0,x) = exp( 1 φ(x)), (3) q(s, x) = p(s, x)exp[ 1 Y t h(x)], where p(s, x) is called pathwise unnormalized filter density. Then, p satisfies the following linear second-order parabolic PDE with coefficients depending on Y T where, for any g R n, let p s = (L s ) p + K p, (4) Lg = 2 tr(g xx)+f g x, K(t, x) = 1 2 (Y t h) x (Y t h) x L(Y t h) 1 2 h 2. (5) L denote the differential generator of the Markov diffusion X t in (1). By assumptions (A1) and (A3) in [12], K is bounded and continuous. Since Y 0 =0,p(0,x)=q(0,x). The initial condition for (4) is given by (3). We rewrite (4) as follows: p s = 1 2 tr(p xx)+a p x + B p, (6) where A = f(x)+(y t h(x)) x, B(t, x) = div[f(x) (Y t h(x)) x ]+K(t, x), (diva x ) j = Σ n i,j=1(a ij ) xi, j = 1,..., n. Taking log transform Z(T,x) = logp(t,x), the nonlinear parabolic PDE is obtained Z s = 2 tr(z xx)+a Z x Z x Z x + B, (7)

3 ICIC EXPRESS LETTERS, VOL.2, NO.4, with initial data Z x (0,x) = φ(x). The risk-sensitive optimal filter problem consists in finding the estimate C T of the state x t, verifying that Z(s, x) = 1(x C 2 s) T Q s (x C s )+ρ s Y t h(x t ), is a viscosity solution of (7). The notation for all the variables is x(t) =x t,x t R n,w t R m,y t R p,f,h R n, where f x,h x are assumed bounded. Here, h x is the matrix of partial derivatives of h. ThesamenotationholdsforZ x. 3. Risk-sensitive Suboptimal Filter. Taking f(x t )=A t +A 1t x t +A 2 (t)x T x+a 3 (t)xxx T, h(x t )=E t + E 1t x t, where A t R n,a 1t, M n n,a 2 (t) is a tensor of dimension n n n, A 3 (t) is a tensor of dimension n n n n, E t R p,e 1t M n p, the following system of stochastic equations is obtained: dx t = A t + A 1t X t + A 2 (t)x T X + A 3 (t)x T XX + d ˆB t, x 0 = x 0, (8) dy t = E t + E 1t X t + d B t. Regarding that f(x t ) is a third degree polynomial and linearizing the expansion of f(x t ) in Taylor series around the equilibrium point ξ, thesuboptimalfiltering algorithms are obtained and presented in the following theorem. Theorem 3.1. The suboptimal solution to the filtering problem for the system (8) with the exponential mean-square criterion (2) takes the form: Ċ s = f 0 (ξ)ξ + f 0 (ξ) T C s Q 1 E 1 (dy E 1t C s E t ), (9) Q s = f 0 (ξ) T Q s Q s f 0 (ξ)+q T s Q s E1tE T 1t, where ξ is the equilibrium point of f(x t ). Proof: The value function is proposed Z(s, X) = 1 2 (X t C s ) T Q s (X t C s )+ρ s Y t (E t + E 1t x t ), where Z x (0,x)= φ(x), C s,q s, ρ s are functions of s [0,T],C s R n,q s is a symmetric matrix of dimension n n and ρ s is a scalar function) as a viscosity solution of the nonlinear parabolic PDE (7). Z x,z xx are the partial derivatives of Z respect to x, and Z is the gradient of Z. The partial derivatives of Z are given by: Z s = 1 2 (X t C s ) T Q s (X t C s )+ ρ s 1 2ĊT s Q s (X t C s ) 1 2 (X t C s ) T Q s Ċ s dy t (E t +E 1t x t ) Z x = 1 2 Q s(x C s )+ 1 2 (X C s) T Q s Y t E 1t, 2 Z x x = Q s. Substituting (10) and the expressions for A, B in (7), the following expression is obtained: 1 2 (X t C s ) T Q s (X t C s )+ ρ s 1 2ĊT s Q s (X t C s ) 1 2 (X t C s ) T Q s Ċ s dy t (E t + E 1t x t )= 2 (q 11 + q q nn )+[ f 0 (ξ)ξ f 0 (ξ)x + Y T Ė 1 ][Q T (x C s ) Y T E 1 ]+ 1 2 [Q T (x C T ) Y T E 1 ][Q T (x C T ) Y T E 1 ] A Y T E 1 Y T E 1 (f 0 (ξ)ξ + f 0 (ξ)x)y T E (E + E 1x) 2 Collecting the second degree terms, equalizing them to zero, and doing it again for the terms of first degree, the risk-sensitive filtering equations (9) are obtained. In similar manner, collecting the zero degree terms, the equation for ρ s is obtained. Here, Q T is a symmetric negative definite matrix, and the initial condition Q 0 = q 0 is derived from initial conditions for Z. If φ(x t )=x T t Kx t, Q(0) = K. It is easy to verify that if Q = P 1, where P is the covariance matrix, those equations are equivalent to the Kalman-Bucy filtering equations, as was shown in [9].

4 374 M. A. ALCORTA, M. BASIN, J. J. MALDONADO AND S. G. ANGUIANO 4. Example. Consider the pendulum equations with friction [14] for the dynamical system with state and output equations: ẋ 1 = x 2, ẋ 2 = g l Senx 1 k m x 2 + ψ 1 (t),x i0 = x io, y t = x 1 + ψ 2 (t), (10) where x R 2,g = 9.8m/seg 2 is the gravity constant, l = 0.5m is the length of the road, m =0.25kg denotes the mass of the bob, θ is the angle subtended by the road and the vertical axes through the pivot point, k = denotes the friction coefficient. White Gaussian noises ψ 1, ψ 2 are the derivatives of the Brownian motion W i, which are independent of each other and the initial condition x i0 = x io,is a varying parameter. Note that x 1 is the measured variable, while x 2 is observable but not measured. Expanding in Taylor series around the origin, this system is reduced to the third degree polynomial form: ẋ 1 = x 2, ẋ 2 = g (x l 1 x3 1 3! ) k x m 2 + ψ 1 (t), x i0 = x i, y t = x 1 + ψ 2 (t). Further linealization yields the linear dynamical system: dx 1t = x 2, dx 2 = g l x 1 k m x 2dt + ψ 1 (t), x i0 = x i, y t = x 1 dt + ψ 2 (t), (11) Substituting the corresponding values from (10) into (9), the equations for the suboptimal risk-sensitive filtering algorithms are given by: Ċ 1 = g l C (q q (ẎT C 1 )+q 12 C 2 ), q 22 q 11 Ċ 2 = C 1 k m C 2 1 q 2 12 q 22 q 11 (q 21 (ẎT C 1 )+q 11 C 2 ), (12) where q 12,q 22,q 11 are the solutions of the following symmetric Riccati matrix equation : q 11 T =2 g l q 21 + q q12 2 1, q 12 T = g l q 22 q 11 + k m q 12 + q 11 q 12 + q 12 q 22, (13) q 22 T = 2k m q 22 2q 21 + q q The last equations (12) and (13) are simulated using Simulink in MatLab7. The initial conditions for the simulation are x 0 =0, q 11 (0) = 8,q 12 (0) = 1.75,q 22 (0) = 1.5, C 1 (0) = 1,C 2 (0) = 1, T =10. The graphs of the difference between the state x t, and the estimate C T,thatis,e i = x i C i, for i =1, 2, with =1000, are shown in Figure 1. Applying the extended Kalman-Bucy filter algorithms [15] to the state equations (11), the equations for the estimate vector m(t) and symmetric covariance matrix P (t) are obtained: dm 1 (t) =m 2 (t)dt + p 11 (dy (t) m 1(t)dt), dm 2 (t) = g l m 1(t)dt k m m 2(t)dt + p 12 (dy (t) m 1 (t)dt), ṗ 11 (t) = 2p 12 (t) p2 11(t)+p 2 12(t) +, ṗ 12 (t) = g l p 11(t) k m p 12(t)+p 22 (t) p 11 (t)p 12 (t)+p 22 (t)p 12 (t), ṗ 22 (t) = 2 g l p 12(t) 2 k m p 22(t) p2 12(t)+p This system of equations is simulated with the initial conditions: m 1,2 (0) = 1, p 11 (0) = ,p 12 (0) = ,p 22 (0) = The graph of the difference between state x i, and the estimate m i (t), that is, e i = x i m i, for =1000, can be observed in

5 ICIC EXPRESS LETTERS, VOL.2, NO.4, Table 1. Comparison of mean-square criterion (2) for R-s filter, third degree polynomial and K-B extended filter for certain values of. value JR S J polynomial Jext.K B Figure 3. The third degree polynomial filtering equations from [1] are given by ṁ 1 (t) =m 2 (t)+ p 11 (Ẏ m 1(t)), ṁ 2 (t) = g l m 1(t) k m m 2(t)+ g 6l (3p 11m 1 (t)+m 3 1(t)) + p 21 (Ẏ m 1(t)), ṗ 11 (t) =2p p2 11, ṗ 12 (t) = g l p 11(t) k (t)+p 22 + g m 12 2l p2 11(t)+ g 2l p 11(t)m 2 1 (t) 2 p 11p 12, ṗ 22 (t) = 2g l p 12(t) 2k (t)+ g m 22 l p 11(t)p 12 (t)+ g l p 12(t)m 2 1 (t). The graph of the difference between state x i, and the third degree polynomial estimate m i (t), that is, e i = x i m i, for =1000, canbeobservedinfigure2. Table1presentssome values of the exponential mean-square cost function corresponding to the risk-sensitive, extended Kalman-Bucy and third degree polynomial filters. It can be observed that the J r s values are less that the J ext.k B and J polynomial values for large values of the parameter. Figure 1. Graphs of the absolute values of the difference between the state x t and the risk-sensitive estimate C T,for = Conclusions. This paper presents the suboptimal solutions to the risk-sensitive optimal filtering problem for stochastic third degree polynomial systems with Gaussian white noises, an exponential-quadratic criterion to be minimized, and intensity parameters multiplying the white noises, using Taylor series for linearizing the third degree polynomial in the state equation. Numerical simulations are conducted to compare performance of the obtained risk-sensitive filter algorithms against third degree polynomial filtering algorithms and extended Kalman-Bucy filter, through comparing the exponential mean-square criteria values. The simulation results reveal strong advantages in favor of the designed

6 376 M. A. ALCORTA, M. BASIN, J. J. MALDONADO AND S. G. ANGUIANO Figure 2. Graphs of the absolute values of the difference between the x t and the third degree polynomial estimate m(t), for = Figure 3. Graphs of the absolute values of the difference between the x t and the Kalman-Bucy extended estimate m(t), for = risk-sensitive suboptimal algorithms in regard to the final criteria values, corresponding to large values of the intensity parameters multiplying diffusion terms in the state and observation equations. Acknowledgment. The first author thanks the UCMEXUS-CONACyT Foundation for financial support under Posdoctoral Research Fellowship Program and Grant The second author thanks the Mexican National Science and Technology Council (CONACyT) for financial support under Grants and REFERENCES [1] M. V. Basin and M. A. Alcorta-Garcia, Optimal filtering and control for third degree polynomial systems, Dynamics of Continuous Discrete and Impulsive Systems, vol.10, pp , [2] M. V. Basin and M. A. Alcorta-Garcia, Optimal filtering for bilinear systems and its application to terpolymerization process state, Proc. of the IFAC 13th Symposium System Identification, pp , [3] M. V. Basin, J. Perez and R. Martinez-Zuniga, Optimal filtering for nonlinear polynomial systems over linear observations with delay, International Journal of Innovative Computing, Information and Control, vol.2, pp , 2006.

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