A Comparison of the Extended and Unscented Kalman Filters for Discrete-Time Systems with Nondifferentiable Dynamics

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1 Proceedings of the 27 American Control Conference Marriott Marquis Hotel at Times Square New Yor City, USA, July -3, 27 FrA7.2 A Comparison of the Extended and Unscented Kalman Filters for Discrete-Time Systems with Nondifferentiable Dynamics J. Chandrasear, A. J. Ridley, and D. S. Bernstein Abstract We compare the performance of the extended Kalman filter, the unscented Kalman filter, and two extensions of the H filter when applied to discrete-time nonlinear state estimation problems with nondifferentiable dynamics. We compare the performance of all the estimation techniques on simple nonlinear examples and finally consider state estimation of one-dimensional hydrodynamic flow based on a finite volume model that contains nondifferentiable nonlinearities. I. INTRODUCTION Because of the widespread need for nonlinear observers and estimators, this area of research remains one of the most active [, 2]. One of the main drivers of research in this area is applications to distributed, large scale systems, the most visible of which is weather forecasting [3]. This area is often referred to as data assimilation. The classical Kalman filter for linear systems is often applied to nonlinear systems in the form of the extended Kalman filter () [4, 5]. A variation of is the statedependent Riccati equation (SDRE) approach, in which, in place of the Jacobians, the dynamics and output map are exactly factored, and the factors are used for the pseudocovariance update [6, 7]. Another approach to state estimation of linear systems are the H filters [8]. Unlie the classical Kalman filter, these filters do not require the stringent Gaussian distribution assumption on the process and sensor noise affecting the system, and guarantee a performance bound. Estimation with uncertainty in the model has also been performed using the H filter [9]. We apply the H filter to nonlinear systems using the Jacobians of the dynamics and measurement maps and call the resulting filter the extended H filter (). Yet another approach to nonlinear estimation involves particle filters. Among the various techniques that have been developed are the unscented Kalman filter () [, ], which deterministically constructs the collection of state estimates. Although particle filters do not require the propagation of a covariance (or pseudo-covariance) in the usual (Riccati) way, the size of the collection determines the computational requirements. Finally, we combine the H filter gain expression with the particle filter framewor to obtain the unscented H filter (). The present paper focuses on discrete-time systems with dynamics that are not differentiable. The main motivation is state estimation based on computational fluid dynamics This research was supported the National Science Foundation, through Grant ATM to the University of Michigan, Ann Arbor, USA. The authors are with the University of Michigan, Ann Arbor, MI-489, dsbaero@umich.edu. (CFD) models for space weather forecasting [2]. In particular we focus on CFD models for hydrodynamics (HD) and magnetohydrodynamics (MHD) in which the equations of fluid motion are approximated finite volume schemes. In [6] we have considered SDRE and methods for state estimation of one-dimensional hydrodynamic flow. In HD and MHD, the CFD models involve nondifferentiable functions as part of the discretization of the underlying partial differential equations [3]. In the present paper, we consider an alternative approach in which we apply and despite the lac of differentiability. In particular, we compute the Jacobian at all points at which it exists, and we employ an averaged value at points at which the dynamics are not differentiable. To demonstrate the accuracy of,,, and when the dynamics are not differentiable, we consider several examples. We are interested in both the accuracy and computational requirements of each approach. II. THE KALMAN FILTER Consider the discrete-time linear system with dynamics and measurements x + = A x + B u + w (2.) y = C x + v, (2.2) where x R n, u R m, and y R p. The input u and output y is assumed to be measured, and w R n and v R p are uncorrelated zero-mean white noise processes with covariances Q and R, respectively. We assume that R is positive definite. For the system (2.) and (2.2), the Kalman filter provides optimal estimates of the state x using measurements y [4]. The Kalman filter equations can be expressed in two steps, namely, the data assimilation step K = P f C T (R f ), (2.3) P da = P f P f CT (Rf ) C P f, (2.4) = x f + K ( y y f ), (data update) (2.5) y f = C x f, (2.6) where R f C P f CT + R, and the forecast step x f + = A + B u, (physics update) (2.7) P+ f = A P da + Q, (2.8) where the data assimilation error covariance P da R n n and the forecast error covariance P f Rn n are defined /7/$ IEEE. 443

2 FrA7.2 P f E [ e f (ef )T], P da E [ e da (eda )T], and the data assimilation error state e da and forecast error state e f are defined e da x e f x x f. Note that the Kalman filter gain K in (2.3) minimizes the cost function J (K ) = tr(p+ f ). III. THE H FILTER Consider the cost function N J(K ) = (ef i )T Me f i (e f )T P fef + N wt i Qw i + N vt i Rv.(3.) i The H filter ensures that inspite of the worst possible process and sensor noise, the cost J(K ) satisfies J(K ) γ. (3.2) The data assimilation step of the robust H filter is given where and = x f + K (y y), f (3.3) y f = Cx f, (3.4) P da = (I K C) P f (I K C) T + K RK T,(3.5) K = P f CT (C P f CT + R) (3.6) P f P f (I γmp f ). (3.7) The forecast step of the H filter is given x f + P f + = A, (3.8) = A P da + Q. (3.9) Note that unlie the Kalman filter, w and v need not be white noise processes and hence Q and R are not their covariances, but a weighting on the uncertainty associated with the process and sensor noise. Moreover, P f da and P in (3.3)-(3.9) are not the error covariances. IV. THE EXTENDED KALMAN FILTER Next, we consider the discrete-time nonlinear system with dynamics and measurements x + = f(x, u, ) + w (4.) The two-step is given y = h(x, ) + v. (4.2) x f + = f(xda, u, ), (4.3) = x f + K (y y), f (4.4) y f = h(xf, ), (4.5) where K, P da and P f are given (2.3), (2.4) and (2.8), respectively, with f(x, u, ) h(x, ) A, C.(4.6) x x x=,u=u x= If f(x, u, ) and h(x, ) are not differentiable with respect to x, in (4.3)-(4.6) cannot be used because A and C defined in (4.6) may not exist for all. However, we assume that the first order symmetric partial derivatives [6] of f(x, u, ) and h(x, ) exist everywhere, that is, for all x R n, and sf(ξ, u, ) sξ i sh(ξ, ) sξ i ξ=x lim δ f(x + δe i, u, ) f(x δe i, u, ) 2δ ξ=x lim δ h(x + δe i, ) h(x δe i, ) 2δ (4.7) (4.8) exist, where ξ R n has scalar entries ξ = [ ξ ξ n ] T and ei R n is the ith column of the n n identity matrix. Hence, for example, although f(x) = x does not have a derivative at x =, it follows from (4.7) that sf sx () =. Furthermore, if g : Rn R is a differentiable function, then the symmetric partial derivative and the partial derivative are equal. Next, we define the (i, j) entry of the averaged Jacobian F s (x, u, ) R n n and H s (x, ) R p n of f( ) and h( ), respectively, sfi(ξ, u, ) shi(ξ, ) F s,i,j(x, u, ), H s,i,j(x, ), (4.9) sξ j ξ=x sξ j ξ=x where f i (x, u, ) and h i (x, ) are the scalar entries of f(x, u, ) R n and h(x, ) R p, respectively. Note that if f( ) and h( ) are differentiable, then, for all x R n, the averaged Jacobians F s and H s are equal to the true Jacobians. Hence, for (4.)-(4.2) when f( ) and h( ) satisfy (4.7) and (4.8) is given (4.3), where K, P da and P f are given (2.3), (2.4) and (2.8), respectively, with A = F s (, u, ), C = H s (, ). (4.) V. THE EXTENDED H FILTER An alternative approach to state estimation of (4.)-(4.2) is based on the H filter. Although, the H filter is derived for linear time-invariant systems, lie the extended Kalman filter, the Jacobian of the dynamics and measurements maps can be used in the filter equations. However, the performance bounds guaranteed in the linear case are not valid anymore. The extended H filter () is given (4.3)-(4.5), where K, P da and P f are given (3.5)-(3.7) and (3.9), with A and C defined (4.). Note that since the Jacobians are based on the symmetric derivatives, that uses the averaged Jacobians can be used on nonlinear systems with nondifferentiable dynamics. Finally, we use γ, Q and R in as tuning parameters to improve the estimates. Note that may not be stable for all values of γ and hence γ must be tuned carefully. VI. THE UNSCENTED KALMAN FILTER Another approach to state estimation of nonlinear systems is the unscented Kalman filter (). The starting point for is a set of sample points, that is, a collection of state estimates that capture the initial probability distribution of the state []. 4432

3 FrA7.2 Assume that x R n, P R n n is positive semidefinite and λ >. The unscented transformation is used to obtain 2n + sample points X i R n and corresponding weights γ x,i and γ P,i, for i =,...,2n, so that the weighted mean and the weighted variance of the sample points is x and P, respectively. The unscented transformation X = Ψ(x, P, λ) of x with covariance P is defined x, if i =, X i = x + P i, if i =,...,n, (6.) x P i n, if i = n +,...,2n, where P ( λp ) /2, for i =,..., n, Pi is the ith column of P, X R n 2n+ has entries X = [ ] X X 2n and λ determines the spread of the sample points around x. Note that 2n γ x,ix i = x, and 2n γ P,i(X i x)(x i x) T = P, where the weights are defined γ x, λ n λ, γ P, λ n λ + ( λ n + β), where β >, and for i =,...,2n, γ x,i = γ P,i 2λ. The analysis step of the unscented Kalman filter is given where = xf + K (y y f ), (6.2) y f = h(x f, ), (6.3) X da, P da, λ), (6.4) P da = P f K P yy, K T, (6.5) K = P xy, P yy,, (6.6) P xy, = P yy, = γ P,i (X f i, xf )(Y f i, yf )T, (6.7) γ P,i (Y f i, y f )(Y f i, y f ) T + R, (6.8) Y f i, = h(x f i,, ), (6.9) and the forecast step of the unscented Kalman filter is given ˆX i,+ f = f(xda i,, ), (6.) x f + = γ x,i ˆXf i,+, (6.) P+ f = γ P,i ( ˆX i,+ f xf + )( ˆX i,+ f xf + )T +Q, (6.2) X f + = Ψ(x f +, P f +, λ). (6.3) Since involves 2n + model update, the computational burden of is of the order (2n+)n 2 = 2n 3 +n 2. On the other hand, involves a single model update and covariance propagation using the Riccati equation and hence the computational burden of is of the order n 3 + n 2. Hence, when n is large the computational burden of is approximately twice that of. The performance of and are compared in []. VII. THE UNSCENTED H FILTER Finally, we consider an extension of that is based on the H filter. The analysis step of the unscented H filter () is given (6.2)-(6.4) with where P da = P f K Pyy, K T, (7.) K = P xy, P yy,, (7.2) P xy, = P yy, = γ P,i ( X f i, x f )(Ỹ f i, y f ) T, (7.3) γ P,i (Ỹ f i, y f )(Ỹ f i, y f ) T + R, (7.4) Ỹi, f = h( X i, f, ), (7.5) and the forecast step of is given (6.)-(6.2), Xf, is obtained using X f + = Ψ(x f +, P f +, λ), (7.6) and P f is defined (3.7). Note that when the dynamics are linear, then is equivalent to presented in Section 3. VIII. EXAMPLES Next, we use,,, and for state estimation of low-dimensional discrete-time systems with nondifferentiable nonlinearities. Specifically, we consider nonlinearities that are not differentiable but have symmetric derivatives everywhere. A. Absolute Value Function First, we consider nonlinearities that commonly occur in finite volume discretization of hyperbolic partial differential equations [3]. For example, the absolute value function appears in the first-order upwind discretization of an advection equation [3]. Let x R 4 and x + = abs(sin(mx )) + w, y = Cx + v, where M R 4 4 and [ C = (8.) ], (8.2) and w and v are zero-mean white processes with covariances Q =.I 4 and R =.I 2, respectively. Note that for all x R, 8 ><, if x >, sabs(ξ) =, if x <, sξ ξ=x >:, if x =. (8.3) Hence, it follows from (4.9), (8.) and (8.3) that for i, j =,...,n, the (i, j) entry of F s (x) is given 8 >< cos(row i(m)x)m i,j, if sin(row i(m)x) >, F s,i,j(x)= cos(row i(m)x)m i,j,if sin(row i(m)x) <, >: 4, if sin(row i(m)x) =, where row i (M) is the ith row of M, and H s (x) = C. (8.4) 4433

4 FrA7.2 Figure shows a plot of abs(sin(mx)) and it can be seen that as m increases, the nonlinearities become more prominent. The logarithm of the sum of the Euclidean norms of the errors in the state estimates for 5 different choices of M with sprad(m) =.5 is shown in Figure 2. Numerical simulations suggest that the performance of,,, and is almost indistinguishable for all choices of M. The error in the state estimates when no data assimilation is performed, that is, K = for in, is also plotted for comparison. Next, the performance of all the estimators for 5 different choices of M with sprad(m) = is shown in Figure 3. It can be seen that in the case of more severe nonlinearities, the performance of and is better than the performance of and. The values of γ in all the cases were chosen such that and are both stable. B. Minmod Function Next, we consider discrete-time systems involving the minmod function, which is used in second-order upwind finite volume schemes as a slope limiter to reduce the diffusion effects [3]. For α, β R, define minmod(α, β) = (sign(β) + sign(β)) min{ α, β } (8.5) 2 (see Figure 4). Let x R and x + =sin(mx )+minmod(m L x, M R x ) + w, y = Cx + v. (8.6) We choose M R so that sprad(m) <, and for i, j =,...,, the (i, j) entry of M L R is given (M L) i,i =, (M L) i,i =, (M L) i,j = if j / {i, i }, (8.7) M R = ML T, and for all, C R 2 is chosen to be [ ] C = 9. (8.8) 9 We assume that w and v are zero-mean white processes with covariances Q =.I and R =.I 2, respectively. Note that for all u, v R, 8, if uv < or u = v =,, if uv > and u > v, >< s minmod(α,, if u, v =, sα β) (u,v) =.5, if uv > and u = v,.5, if u =, v, >:, if uv > and u < v. (8.9) Furthermore, (8.6) implies that H s (x) = C. The sum of the Euclidean norm of the error in the state estimates obtained from,,, and for 5 different choices of M with sprad(m) =.5, is shown in Figure 5. The performance of the four estimators for 5 different choices of M with sprad(m) =. is shown in Figure 6. IX. SIMULATION EXAMPLE : ONE-DIMENSIONAL HYDRODYNAMICS Finally, we consider state estimation of one-dimensional hydrodynamic flow based on a finite volume model. The flow of an inviscid, compressible fluid along a one-dimensional channel is governed Euler s equations. A discrete-time model of hydrodynamic flow can be obtained using a finite-volume based spatial and temporal discretization. Assume that the channel consists of n identical cells. For all i =...,n, define U [i] R 3 U [i] = [ ] ρ [i] T, m [i] E [i] where ρ [i] is the density, m [i] is the the momentum and E [i] is the energy in the ith cell. We use a second-order Rusanov scheme [3] to discretize Euler s equations and obtain a discrete-time model that enables us to update the flow variables at the center of each cell. Define the state vector x R 3(n 4) [ x = (U [3] )T (U [n 2] ) T ] T. (9.) For all, let u R 3 denote the boundary condition for the first two cells, so that u = (U [] )T = (U [2] )T. Furthermore, we assume Neumann boundary conditions at cells with indices n and n. The second-order Rusanov scheme yields a nonlinear discrete-time update model of the form (4.), where w R 3(n 4) represents unmodeled drivers and is assumed to be zero-mean white Gaussian process noise with covariance matrix Q R 3(n 4) 3(n 4) such that only the flow variables in the th, 25th and 4th cell are directly affected w. We assume that measurement y R 6 of density, momentum and energy at cells with indices 6, 6, 26, 35, and 42 are available and and v is zero-mean white Gaussian noise with covariance matrix R =.I 5 5. Let n = 54 so that x R 5. For all, let [] = [2] = g/m 3, m [] = m [2] = v in + vin 4 sin() m/s, and SE [] = E [2] = (/2)(m [] )2/ [] + 3/2 N/m2, where v in is the inlet velocity. We simulate the truth model from an arbitrary initial condition x R 3(n 4) and obtain measurements y for various choices of v in {.,., 2.,...,.} m/s. Note that if v in >.29 m/s, then the flow is supersonic. The objective is to estimate the density, momentum, and energy at the cells where measurements of flow variables are unavailable using and. It follows from (3.7) that and involve inverting a n n matrix which is computationally intensive when n is large which is the case in finite volume discretization of partial differential equations. Moreover, in the previous examples, no significant improvement in performance was noticed when the and were used instead of and, respectively. Hence, we do not use or for state estimation in the one-dimensional hydrodynamic flow example. The error in the estimates of the energy E [3] in cell 3, when measurements y are used in and with v in = m/s is shown in Figure 7. The error in estimates of the energy E [3] in cell 3, when v in = m/s is shown in Figure 8. The sum of the Euclidean norm of error in the state 4434

5 FrA7.2 estimates for different values of v in is shown in Figure 9. Note that at low inlet velocities v in, the performance of and is very similar. However, at higher inlet velocities, the nonlinearities are more severe and the performance of is better than that of. X. CONCLUSION In this paper we compared the performance of the extended Kalman filter, the extended H filter, the unscented Kalman filter, and the unscented H filter for nonlinear systems with nondifferentiable nonlinearities. Whenever the Jacobian fails to exist, we use an averaged Jacobian based on the symmetric derivatives in the extended Kalman filter. For all the examples that we considered, whenever the nonlinearities are not severe, the performance of with the averaged Jacobian and is similar. However, when the nonlinearities become severe, performs better that. No significant improvement in the performance was noticed when either the extended H filter or the unscented H filter was used over the extended Kalman filter and unscented Kalman filter, respectively. REFERENCES [] M. Athans, R. P. Wishner, and A. Bertolini, Suboptimal State Estimation for Continuous-Time Nonlinear Systems from Discrete Noisy Measurements, IEEE Trans. Auto. Ctrl., vol. 3, pp , 968. [2] K. Ito and K. Xiong, Gaussian Filters for Nonlinear Filtering Problems, IEEE Trans. Auto. Ctrl., vol. 45, pp , 2. [3] J. M. Lewis, S. Lashmivarahan, and S. Dhall, Dynamic Data Assimilation : A Least Squares Approach. Cambridge University Press, 26. [4] A. Jazwinsi, Stochastic Processes and Filtering Theory. Academic Press, 97. [5] A. Gelb, Applied Optimal Estimation. Cambridge: MIT Press, 974. [6] J. Chandrasear, A. J. Ridley, and D. S. Bernstein, A SDRE-Based Asymptotic Observer for Nonlinear Discrete-Time Systems, in Proc. Amer. Contr. Conf., Portland, OR, June 25, pp [7] C. P. Mrace, J. R. Cloutier, and C. A. D Souza, A New Technique for Nonlinear Estimation, in Proc. Int. Conf. Contr. App., Dearborn, MI, June 996, pp [8] W. Sun, K. M. Nagpal, and P. P. Khargonear, H Control and Filtering for Sampled-Data Systems, IEEE Trans. Auto. Ctrl., vol. 38, pp , 993. [9] E. G. Collins Jr. and T. Song, Robust H Estimation and Fault Detection of Uncertain Dynamic Systems, Int. J. Guid. Cont. Dyn., vol. 23, no. 5, pp , 2. [] S. Julier, J. Uhlmann, and H. F. Durrant-Whyte, A New Method for the Nonlinear Transformation of Means and Covariances in Filters and Estimators, IEEE Trans. Auto. Ctrl., vol. 45, pp , 2. [] R. V. der Merwe and E. A. Wan, The Square-root Unscented Kalman Filter for State and Parameter-Estimation, in Proc. Int. Conf. Acou. Speech Sig. Process., May 2, pp [2] C. Groth, D. D. Zeeuw, T. Gombosi, and K. Powell, Global 3D MHD Simulation of a Space Weather Event: CME Formation, Interplanetary Propagation, and Interaction with the Magnetosphere, J. Geophys. Res., vol. 5, pp , 2. [3] C. Hirsch, Numerical Computation of Internal and External Flows. John Wiley and Sons, 99. [4] B. D. O. Anderson and J. B. Moore, Optimal Filtering, Dover Publications Inc., Mineola, NY, 979. [5] L. Scherliess, R. W. Schun, J. J. Soja, and D. C. Thompson, Development of a Physics-based Reduced State Kalman filter for the Ionosphere, Radio Science, vol. 39, 24. [6] L. Larson, The Symmetric Derivative, Trans. Amer. Math. Soc., vol. 277, pp , 983. log (Σ e /2 ).5.5 abs(sin(mx)).5.5 m=.5 m= x Fig.. Plot of abs(sin(mx)) for m =.5 and m = Fig. 2. Logarithm of the sum of Euclidean norms of the errors in state estimates obtained using,,, and for the system (8.). The performance is compared for 5 different choices of M with sprad(m) =.5. The error in the estimates when no data assimilation is performed, that is, K = for all in is also shown for comparison. log (Σ e /2 ) Fig. 3. Logarithm of the sum of the Euclidean norms of the errors in state estimates obtained using,,, and for the system (8.). The performance is compared for 5 different choices of M with sprad(m) =. The performance of and is much better than the performance of or. However, the performances of and are very similar to the performance of and, respectively. 4435

6 FrA Two step.3 Error in energy estimates at cell Fig. 4. Plot of minmod(α, β) for 5 α, β < time in s.5.95 Fig. 7. The error in the estimates of energy at cell 3 obtained using and when v in = m/s and the flow is subsonic..8 Two step log (Σ e /2 ) Error in energy estimates at cell Fig. 5. Logarithm of the sum of the Euclidean norms of the errors in state estimates obtained using,,, and for the system (8.6). The performance of the four estimators are compared for different choices of M with sprad(m) =.5. We choose the largest possible γ (=.5) such that both and are stable for all choices of M time in s Fig. 8. The error in the estimates of velocity at cell 3 obtained using and when v in = m/s and the flow is supersonic with Mach number Two step log (Σ e /2 ).5 Σ e Fig. 6. Logarithm of the sum of the Euclidean norms of the errors in state estimates obtained using,,, and for the system (8.6). The performance of the two estimators is compared for 5 different choices of M with sprad(m) =.. There seems to be no significant improvement in the performance when the H filters ( and ) are used over and, respectively v in m/s in Fig. 9. The square root of the sum of the Euclidean norms of the errors in state estimates, obtained using and for different choices of the inlet velocity v in. The performance of is better that the performance of for high inlet velocities, with a computational burden that is twice that of. 4436