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Desensitized cubature Kalman filter with uncertain parameters
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Desensitized cubature Kalman filter with uncertain parameters Tai-Shan Lou a, Lei Wang b,∗, Housheng Su c, Mao-Wen Nie d, Ning Yang a, Yanfeng Wang a a School
of Electrical and Information Engineering, Zhengzhou University of Light Industry, Zhengzhou 45002, China b School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China c School of Automation, Image Processing and Intelligent Control Key Laboratory of Education Ministry of China, Huazhong University of Science and Technology, Wuhan 430074, China d Institute of High Performance Computing, A∗ STAR, Singapore, 138632, Singapore Received 25 October 2016; received in revised form 6 August 2017; accepted 1 September 2017 Available online xxx
Abstract In this study, a robust desensitized cubature Kalman filtering (DCKF) is proposed for nonlinear systems with uncertain parameters. Unlike the cubature Kalman filtering, the desensitized cost function is introduced by penalizing the posterior covariance trace with a weighted sum of the posteriori sensitivities. The sensitivity of the root square matrix is obtained by solving a Lyapunov-like linear equation, and the sensitivity propagation of the state estimate errors is presented. The effectiveness of the proposed DCKF is demonstrated by two numerical examples in which models with uncertain parameters are considered. © 2017 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction Nonlinear state estimation plays an important role in a wide variety of applications, including estimation in navigation of aerospace vehicles [1], state of the chemical plant control [2], position and velocity of target tracking [3], and fusion of multi-sensor data [4]. A great deal of estimation methods such as extended Kalman filter (EKF), unscented Kalman filter (UKF), ∗
Corresponding author. E-mail addresses: [email protected], [email protected], [email protected] (L. Wang).
https://doi.org/10.1016/j.jfranklin.2017.09.004 0016-0032/© 2017 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
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divided difference filter (DDF), and particle filter have been proposed to solve different practical problems. However, the existing nonlinear state estimation methods still suffer from high computational cost. For instance, the EKF is computationally intensive due to unstable calculation of Jacobians or Hessians [5–7]; the UKF and DDF may not converge or converge with high computational cost [8]; the particle filter suffers from expensive computational load [9]. Recently, a nonlinear filter, called cubature Kalman filter (CKF), for the Gaussian systems has been proposed based on the Bayesian framework [10,11]. The core idea of the CKF is using the cubature role to approximate the multidimensional integrals with 2n deterministic cubature points, where n is the state vector dimension. In the CKF algorithm, the mean and the covariance of the state and the measurement vectors are calculated by propagating the cubature points through the nonlinear function. Compared with the EKF and the UKF, the CKF is usually applied in high-dimensional systems since it can produce higher accuracy and take less computational cost [10]. Subsequently, the CKF has been used in many applications including navigation [1], attitude estimation [12], chemical processing [13], and sensor data fusion [14,15], etc. As a matter of fact, all above-mentioned filters assume that the parameters of the dynamic model are accurately known as a prior and the noises of the dynamic model are white with normal probability distributions [5]. It has been shown that the performances of these filters is highly sensitive to the uncertainties of the dynamic model, and deteriorate appreciably [16]. To solve this issue, Karlgaard and Shen extended a desensitized optimal control methodology [17], and proposed a robust filter to reduce the performance sensitivity of the filters with respect to uncertain model parameters. The desensitized Kalman filter is designed by introducing a weighted norm of the state error sensitivities to the cost function, and minimizing this cost function to obtain the desensitized state estimates [17]. The desensitized divided difference filtering [18] and the desensitized unscented Kalman filtering (DUKF) [19] were presented. For the DUKF, there is no continuous propagation for the sensitivities of the sigma points between the iterations, because a new set of sigma points is always resampled at the next iteration. Shen and Karlgaard skillfully designed a unique way to propagate the sensitivities of the state estimate errors and the priori/posteriori covariance matrices for the DUKF [19]. But for the CKF, the desensitized optimal control methodology is not introduced, and the sensitivity of the root square matrix in filtering should be redesigned to avoid the calculations of the vectorized formula and Kronecker product. This paper proposes a robust cubature Kalman filter for dynamic systems with uncertain parameters. A desensitized cost function of the robust desensitized cubature Kalman filtering (DCKF) is designed by penalizing the posteriori covariance trace with a sensitivity-weighting sum of the posteriori sensitivities. Then, a gain matrix of the DCKF is obtained by minimizing the new cost function to amend the state estimation. The rest of this paper is organized as follows: Section 2 briefly introduces the recursive Bayesian framework and the CKF algorithm. Section 3 presents the propagation of the sensitivities and the DCKF is proposed naturally. Two numerical simulations about a vertically falling body model and a hovering helicopter model (HHM) are analyzed in Section 4. Conclusions are drawn in the end. 2. Cubature Kalman filter Consider a discrete nonlinear system model with additive noises, which is given by xk = f (xk−1 , c, uk−1 ) + wk−1 Please cite this article as: T.-S. Lou et al., Desensitized filter with uncertain parameters, Journal of the Franklin https://doi.org/10.1016/j.jfranklin.2017.09.004
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zk = h(xk , c, uk ) + vk
(2)
where xk ∈ Rn is the state vector at time step k, and zk ∈ Rm is the measurement vector at time step k, vector-valued functions f and h describe the nonlinear dynamic equation and the measurement equation, respectively, c ∈ R is referred to as the uncertain parameter vector, and uk ∈ Rn is the known control input vector. wk and vk are independent zero-mean Gaussian noise processes satisfying E wk wTj = Qk δi j , E vk vTj = Rk δi j , E wk vTj = 0 (3) where δ kj is the Kronecker delta function, Qk ∈ Rn×n is a positive semi-definite matrix, and Rk ∈ Rm×m is a positive definite matrix. 2.1. Recursive Bayesian filter under Gaussian domain It is easy to verify that under the Gaussian domain, the predictive density p(xk |Zk−1 ), the filter likelihood density p(zk |Zk .) and the a posterior density p(xk |Zk .) are all Gaussian, where Zk denotes the historic input measurement pairs {ui , zi }ki=1 up to time tk . Assume that + + the posterior density p(xk−1 |Zk−1 ) is N (xˆ k−1 , Pk−1 ) at time tk−1 . Then, some prerequisite distributions in filter are respectively given by p(xk |Zk−1 ) = N xˆ k− , Pk− p(zk |Zk−1 ) = N zˆ k− , Pz−z,k (4) p(xk |Zk ) = N xˆ k+ , Pk+ where N(·,·) is the Gaussian density symbol, the superscripts “− ” denotes a priori and “+ ”denotes a posteriori. xˆ k− , xˆ k+ , and zˆ k− are the state estimates and the measurement estimate, respectively; Pk− , Pk+ , and Pz−z,k are the corresponding error covariance matrices. Based on the above discussion, the functional recursion of the Bayesian filter reduces to an algebraic recursion, i.e. only the means and the covariance in Eq. (4), need to be updated at each time step. The recursive Bayesian filter under the Gaussian approximation is then summarized as follows [10]: Time update + − + dxk−1 xˆ k = f (xk−1 , c¯ , uk−1 )N xˆ k−1 , Pk−1 (5) Rn
Pk− =
Rn
+ + dxk−1 − xˆ k− xˆ k−T + Qk−1 f (xk−1 , c¯ , uk−1 )f T (xk−1 , c¯ , uk−1 )N xˆ k−1 , Pk−1
(6)
where x˜ k− = xˆ k− − xk is the priori estimation error, and c¯ is the reference value of the parameter vector c. Measurement update zˆ k− = h (xk , c¯ , uk )N xˆ k− , Pk− dxk (7) Rn
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− Pzz, k = − Pxz,k =
Rn
Rn
h (xk , c¯ , uk )hT (xk , c¯ , uk )N xˆ k− , Pk− dxk−1 − zˆ k− zˆ k−T + Rk
(8)
xk hT (xk , c, uk )N (xˆ k− , Pk− )dxk−1 − xˆ k− zˆ k−T
(9)
xˆ k+ = xˆ k− + Kk [zk − zˆ k− ]
(10)
− −T Pk+ = Pk− − Pxz,k KkT − Kk Pxz,k + Kk Pz−z,k KkT
(11)
−1
− Kk = Pxz,k (Pz−z,k )
(12)
where x˜ k+ = xˆ k+ − xk is the posteriori estimation error. Note that the Kalman optimal gain Kk in Eq. (12) can be solved by minimizing the cost function J = T r(Pk+ ), where “Tr” denotes the trace of the matrix. The detailed proof is omitted here since it can easily deduced by differentiating the covariance matrices J with respect to Kk [20]. 2.2. Cubature Kalman filter algorithm The CKF is a type of the Bayesian filter under the Gaussian assumption, i.e. the dynamic noises and the measurement noises are assumed to be zero-mean Gaussian noises. Then, the CKF approximates the mean and the covariance of a random variable propagated under a nonlinear function by the cubature rule. Under the Gaussian assumption, the functional recursion of the Bayesian filter reduces to an algebraic recursion, in which the multidimensional integrals can be approximated by the cubature rule [10]. The cubature rule approximates the n-dimensional Gaussian weighted integral via the mean μ and the covariance P, which can be formulated as
√ 1 f (x)N (μ, P)dx ≈ f μ + P ξi 2n i=1 Rn 2n
(13)
√ √ √ where P is a square-root factor of the covariance P satisfying P= P ( P )T ; ξ√i is the ith element of 2n cubature points, and its value comes from the set n{e1 , e2 , . . . , en , −e1 , −e2 , . . . , −en}, where ei (i = 1, 2, . . . , n) is the unit vector, in which the ith element is one and others are zeros, i.e. ei =[0, . . . ,0,1,0, . . . ,0]T . + + Based on the state estimate xˆ k−1 and the covariance Pk−1 at time tk−1 , the CKF is given by
+ + χ+ = Pk−1 ξ j + xˆ k−1 , j = 1, 2, . . . , 2n (14) j,k−1 + , c¯ , u χ , j = 1, 2, . . . , 2n χ ∗− = f k−1 j,k−1 j,k
(15)
1 ∗− χ 2n j=1 j,k 2n
xˆ k− =
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1 ∗− ∗−T χ χ − xˆ k− xˆ k−T + Qk−1 2n j=1 j,k j,k
= Pk− ξ j + xˆ k− , j = 1, 2, . . . , 2n 2n
Pk− =
(17)
χ− j,k
(18)
¯ , uk ) Z−j,k = h(χ − j,k , c
(19)
1 − Z 2n j=1 j,k 2n
zˆ k− =
(20)
Pz−z,k =
1 − −T Z Z − zˆ k− zˆ k−T + Rk 2n j=1 j,k j,k
(21)
− Pxz,k =
1 − −T χ Z − xˆ k− zˆ k−T 2n j=1 j,k j,k
(22)
2n
2n
xˆ k+ = xˆ k− + Kk (zk − zˆ k− )
(23)
− −T Pk+ = Pk− − Pxz,k KkT − Kk Pxz,k + Kk Pz−z,k KkT
(24)
− Kk = Pxz,k (Pz−z,k )−1
(25)
3. Desensitized cubature Kalman filter In this section, we present a robust DCKF based on the sensitivity propagation equation and the CKF method. Section 3.1 gives the sensitivity propagation equation for the recursive Bayesian filter, and the sensitivity propagation of cubature points in the CKF is proposed in Section 3.2. 3.1. Sensitivity propagation equation for the recursive Bayesian filter Under the optimality conditions of the recursive Bayesian filter such as no model and parameter uncertainties, the state estimate are unbiased. That is to say, the estimation errors satisfy E x˜ k− = 0, E x˜ k+ = 0 (26) When the optimality conditions do not hold true, the state estimates may be biased and even divergence. So, the sensitivity of the state estimation error to each parameter, ci , can be defined as [17] ∂ x˜ k− ∂ xˆ − = k (27) ∂ci ∂ci ∂ x˜ + ∂ xˆ + + si,k = k = k (28) ∂ci ∂ci where ci denotes the ith component of the parameter vector c. Note that the sensitivity of the true state is ∂x/∂ci = 0 in Eqs. (27) and (28) since the true state does not vary with the assumed value of the parameter ci . − si,k =
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Then, the propagation equations of the recursive Bayesian filter are defined as ∂f (xk−1 , c, uk−1 ) ∂f (xk−1 , c, uk−1 ) + − + = dxk−1 si,k =E N xˆ k−1 , Pk−1 ∂ci ∂ci Rn + − si,k = si,k − Kk γi,k
(29) (30)
where the sensitivity of the nonlinear measurement function is defined by ∂h(xk , c, uk ) ∂h(xk , c, uk ) − − = γi,k = E N xˆ k , Pk dxk ∂ci ∂ci Rn
(31)
Note that the sensitivity of the gain matrix should be satisfied with ∂K/∂ci = 0 in Eq. (30). One key reason is that if ∂ K/∂ ci = 0, then the optimal gain is a function of the residual violating the basis for the Lyapunov-like linear update equation given in Eq. (11) [17]. 3.2. Sensitivity propagation of cubature points According to the partial derivative of the cubature point generated equation, the propagation of the sensitivities is determined bases on the sensitivity of cubature points, such as Eqs. (14) and (18). Then, the propagation equations of the sensitivities can be obtained by taking partial derivative of the propagation equations of the CKF, such as Eqs. (15)–(17) and (19)– (24). The sensitivity propagation algorithm of the cubature points is summarized as following: First, compute the sensitivities of step k − 1 cubature points (i = 1, 2, . . . , , j = 1, 2, . . . , 2n)
+ ∂ Pk−1 ∂χ + j,k−1 + = ξ j + si,k−1 , (32) ∂ci ∂ci and update the sensitivity cubature points + , c¯ , u + ¯ χ c ∂f χ + , , u ∂f k−1 k−1 ∂χ ∗− ∂χ j,k−1 j,k−1 j,k−1 j,k = + ∂ci ∂ci ∂ci ∂χ +
(33)
j,k−1
Second, evaluate the sensitivities of the prior state estimate and the prior covariance matrix ∗−
∂ xˆ k− 1 ∂χ j,k = ∂ci 2n j=1 ∂ci ⎧ ∗− T ⎫ ∗− 2n ⎨ ⎬ − ∂χ ∂χ j,k ∂Pk 1 j,k ∗−T − −T −T = χ j,k + χ ∗− − si,k xˆ k − xˆ k− si,k j,k ⎭ ∂ci 2n j=1 ⎩ ∂ci ∂ci 2n
− si,k =
(34)
(35)
Third, compute the sensitivities of the redrawn cubature points and the predicted measurement cubature points
∂ Pk− ∂χ − j,k − = ξi + si,k , (36) ∂ci ∂ci ¯ , uk ) ∂χ − ¯ , uk ) ∂Z−j,k ∂h(χ − ∂h(χ − j,k , c j,k j,k , c = + , (37) − ∂ci ∂ci ∂ci ∂χ j,k Please cite this article as: T.-S. Lou et al., Desensitized filter with uncertain parameters, Journal of the Franklin https://doi.org/10.1016/j.jfranklin.2017.09.004
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and evaluate the sensitivities of the predicted measurement −
∂ zˆ k− 1 ∂Z j,k = ∂ci 2n j=1 ∂ci 2n
γi,k =
(38)
Fourth, evaluate the innovation covariance matrix and the cross-covariance matrix ⎧ − T ⎫ − 2n ∂Z j,k ⎬ ∂Pz−z,k 1 ⎨ ∂Z j,k −T T = Z j,k + Z−j,k − γi,k zˆ k−T − zˆ k− γi,k ⎭ ∂ci 2n j=1 ⎩ ∂ci ∂ci ⎧ − T ⎫ − 2n − ∂Z j,k ⎬ ∂Pxz,k 1 ⎨ ∂χ j,k −T − −T T = Z j,k + χ − − si,k zˆ k − xˆ k− γi,k j,k ⎩ ∂ci ⎭ ∂ci 2n ∂ci
(39)
(40)
j=1
Lastly, evaluate the sensitivities of the posterior state estimate and the posterior covariance matrix + − si,k = si,k − Kk γi,k
(41)
− −T ∂Pxz,k ∂Pxz,k ∂Pz−z,k T ∂Pk+ ∂Pk− = − KkT − Kk + Kk Kk (42) ∂ci ∂ci ∂ci ∂ci ∂ci √ The sensitivity of the root square matrix P in Eqs. (32) and (36) must be computed when the sensitivities of the cubature points √ √ and redrawn cubature points are calculated. By taking partial derivative of equation P= P ( P )T , which is different from the vectorized formula in Karlgaard and Shen [17], we have √ √ ∂P ∂ P √ T √ ∂ P T = ( P ) + P( ) (43) ∂ci ∂ci ∂ci √ The solution ∂ P/∂ci in Eq. (43) can be obtained as [21] −1 √ T ∂ P ∂P −1 1 T = −
T (44) ∂ci 2 ∂ci
where ∈ Rn×n is a √ skew symmetric matrix such that T = − , and are non-singular matrices satisfying P = I . 3.3. Desensitized cubature Kalman filter algorithm In addition to propagate the sensitivity of cubature points in Section 3.2 and compute the gain matrix different from the CKF, other steps of the proposed DCKF are the likely same as the CKF. So, only the gain matrix is derived in this section. With the sensitivities of the posterior state estimate in Section 3.2, a desensitized cost function of the DCKF consisting of the posterior covariance and a weighted norm of the posterior sensitivity is penalized by + a sensitivity-weighting sum of the sensitivities si,k Jd = T r(Pk+ ) +
+T + si,k Wi,k si,k
(45)
i=1
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Table 1 DCKF algorithm. Step 0: Set k = 0, and initialize the state vector, xˆ 0 , the auxiliary matrix, P0 , and the sensitivity parameters, s0 and ∂P0 /∂ci , i = 1, 2, . . . , ; ∗− Step 1: Evaluate the propagated cubature points, χ ∗− j,k , using (14), and the sensitivities, ∂χ j,k /∂ci , by Eqs. (44) and (32); − Step 2: Estimate the predicted state, xˆ k− , by Eq. (15), and the prior sensitivity, si,k , by Eq. (33); Step 3: Evaluate the propagated cubature points, χ ∗− , by Eq. (16) , and the sensitivities, ∂χ ∗− j,k j,k /∂ci , by Eq. (34); − Step 4: Estimate the predicted covariance matrix, Pk , by Eq. (17), and the sensitivity, ∂Pk− /∂ci , by Eq. (35); + Step 5: Evaluate the redrawn cubature points, χ − j,k , by Eq. (18), and the sensitivities, ∂χ j,k−1 /∂ci , by Eqs. (44) and (36); Step 6: Evaluate the propagated cubature points of measurement equation, Z−j,k , by Eq. (19), and the sensitivities, ∂Z−j,k /∂ci , by Eq. (37); Step 7: Estimate the predicted measurement, zˆ k− , by Eq. (20), and the sensitivity, γ i, k , by Eq. (38); Step 8: Estimate the innovation covariance matrix, Pz−z,k , by Eq. (21), and the sensitivity, ∂Pz−z,k /∂ci , by Eq. (39); − − Step 9: Estimate the cross-covariance matrix, Pxz,k , by Eq. (22), and the sensitivity, ∂Pxz,k /∂ci , by Eq. (40); Step 10: Obtain the gain matrix, Kk , by solving Eq. (46); + Step 11: Estimate the posterior state, xˆ k+ , by Eq. (23), and the posterior sensitivity, si,k , by Eq. (41); + Step 12: Estimate the posterior covariance matrix, Pk , by Eq. (24), and the sensitivity, ∂Pk+ /∂ci , by Eq. (42); Step 13: Set k = k + 1 and return to step 1.
where Wi,k ∈ Rn×n is a symmetric positive semi-definite weighting matrix for the ith sensitivity. Substituting Eqs. (24) and (41) into Eq. (45), taking partial derivative of Jd with respected to the gain matrix Kk , and setting the partial derivative ∂Jd /∂Kk = 0 gives the solution Kk Pz−z,k
+
T Wi,k Kk γi,k γi,k
i=1
=
− Pxz,k
+
− T Wi,k si,k γi,k
(46)
i=1
where the gain Kk is obtained by algebraically solving with the Lyapunov-like linear equation in Eq. (46). The DCKF algorithm is summarized in Table 1. 4. Numerical simulation To demonstrate the effectiveness of the proposed DCKF algorithm in the previous section, the vertical falling body model with one uncertain parameter [20,22] and the HHM with two uncertain parameters are considered [23]. The perfect CKF (perf. CKF) and the imperfect CKF (imp. CKF) are employed to compare the performance of the proposed algorithm. The “perfect” means that the true values of the parameters are all known exactly in perf. CKF; the “imperfect” means that in the imp. CKF the true values of the uncertain parameters are not known, and only the reference values, coming from the previous experience, are known. 4.1. Falling body model with one uncertain parameter In this example, we use the perf. CKF, the imp. CKF and the proposed DCKF to estimate the altitude x1 (t), the velocity x2 (t) and the constant ballistic coefficient x3 (t) of a vertically falling body with a high velocity [20,22]. A tracking radar device is used to record the range measurements between the radar and the falling body. The system equation is given by Please cite this article as: T.-S. Lou et al., Desensitized filter with uncertain parameters, Journal of the Franklin https://doi.org/10.1016/j.jfranklin.2017.09.004
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Fig. 1. Sensitivities of the imperfect CKF and the DCKF for state x1 .
Fig. 2. Sensitivities of the imperfect CKF and the DCKF for state x2 .
Fig. 3. Sensitivities of the imperfect CKF and the DCKF for state x3 .
x˙1 (t ) = x2 (t ) + w1 (t ) x˙1 (t ) = x22 (t )x3 (t ) exp{−x1 (t )/c} − g + w2 (t ) x˙3 (t ) = w3 (t )
(47)
where constant c reflects relationship between the air density and the altitude with its reference value c¯ = 2 × 104 , g = 32.2 ft/s2 is the gravitational acceleration. wi (t ) is the process Please cite this article as: T.-S. Lou et al., Desensitized filter with uncertain parameters, Journal of the Franklin https://doi.org/10.1016/j.jfranklin.2017.09.004
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Fig. 4. x1 RMSE.
Fig. 5. x2 RMSE.
Fig. 6. x3 RMSE.
noise that affects the ith equation with E [wi2 (t )] = 0 (i = 1, 2, 3). The discrete-time range measurement from the radar is given by zk =
M 2 + (x1,k − H )2 + vk
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Fig. 7. True values of the HHM and estimates of the imp. CKF and the DCKF.
where M = 105 ft is the radar horizontal distance from the body’s vertical line of fall, H = 105 ft is the radar altitude above the ground level, and vk is the measurement noise assumed to be a zero-mean Gaussian noise with covariance E [vk2 ] = R = 104 ft2 . Also, assume that the constant parameter c is uncertain, and the true values of c follows a uniform distribution, c ∼ U (3/4c¯, 5/4c¯). The true state and initial estimates and the initial covariance are given as x1 (0) = 3 × 105 ft; x2 (0) = −2 × 104 ft/s; x3 (0) = 1 × 10−3
(49)
xˆ1 (0) = 3 × 105 ft; xˆ2 (0) = −2 × 104 ft/s; xˆ3 (0) = 3 × 10−5
(50)
P0+ = diag 1 × 106 ft2 , 4 × 106 ft2 /s2 , 1 × 10−4
(51)
The total simulation time is fixed at 60 s, and the fourth-order Runge–Kutta method is used to discretize the continuous state equation with a sample frequency 10 Hz. 200 Monte Carlos are run to evaluate the performance of the three filters. In the imp. CKF and the DCKF, the uncertain parameter c is set as the corresponding reference value c¯ = 2 × 104 . For the DCKF, the sensitivity-weighting matrix is set to (52) W = diag 3 × 104 , 6 × 103 , 1 × 105 The state sensitivities with respect to the uncertain parameter c with logarithmic scales are shown in Figs. 1–3. Compared to the imp. CKF, the proposed DCKF almost has smaller sensitivities to the uncertain parameter. Figs. 4–6 show that the root mean squared errors Please cite this article as: T.-S. Lou et al., Desensitized filter with uncertain parameters, Journal of the Franklin https://doi.org/10.1016/j.jfranklin.2017.09.004
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Fig. 8. Sensitivities of the imp. CKF and the DCKF.
(RMSEs) of the above three filters. When the process model is disturbed by the uncertain parameter, the RMSE for the DCKF is smaller than that of the imperfect CKF. In a word, the proposed DCKF can reduce the sensitivities of the state estimate errors respect to the uncertain parameter compared with the imperfect CKF, and has a better performance than the imperfect CKF when the process model parameter is imperfect. 4.2. Hovering helicopter model with two uncertain parameters In this example, the ⎡ c1 c2 ⎢1.26 −1.765 x˙ = ⎢ ⎣ 0 1 1 0
hovering helicopter model is given by [23] ⎤ ⎤ ⎡ −g 0 0.086 ⎢ ⎥ 0 0⎥ ⎥x − ⎢−7.408⎥Klqr x ⎣ 0 ⎦ 0 0⎦ 0 0 0
(53)
zk = xk + vk
(54)
where the state vector is x = [x1 , x2 , x3 , x4 ]T , in which x1 is the horizontal velocity, x2 is the pitch angle of the fuselage and its derivative x3 , and x4 is perturbation from a ground point reference. g is the acceleration from gravity and its value is g = 0.322, Klqr is a constant vector given by Klqr = [1.989, −0.256, −0.7589, 1]. vk is the measurement noise. The two uncertain model parameters c1 and c2 are assumed to be uniform distributions, which are Please cite this article as: T.-S. Lou et al., Desensitized filter with uncertain parameters, Journal of the Franklin https://doi.org/10.1016/j.jfranklin.2017.09.004
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Fig. 9. Mean cost function.
respectively c1 ∼ U (−0.15, −0.05) and c2 ∼ U(0.05, 0.15). The initial true state of the HHM is x0 = [0.7929, −0.0466, −0.1871, 0.5780]T
(55)
and the initial state of the filter is set as xˆ 0 = x0 , with the initial covariance Pˆ 0 = I . In the simulation, the total time interval is assumed to be 4s, and the continuous state equation has been discretized using the fourth-order Runge–Kutta method with a sample frequency 20 Hz. For performance evaluation, 200 Monte Carlo runs are performed for three filters. In the imp. CKF and the DCKF the two uncertain parameters are set as their reference values, which are c¯1 = −0.1 and c¯2 = 0.1. For the DCKF, the sensitivity-weighting matrices are set to (56) W1 = W2 = diag 3 × 10−3 , 2 × 10−3 , 1 × 10−2 , 2 × 10−2 The true state of the HHM and its estimates of the imp. CKF and the DCKF in one Monte Carlo test are shown in Fig. 7. The true values of the two parameters in this simulation case are c¯1 = −0.0729 and c¯2 = 0.1142, respectively. Fig. 8 represents the sensitivity of the state variables to the parameters, c1 , in the imp. CKF and the DCKF with logarithmic scales, because the state sensitivities to c2 are similar. From Fig. 8, it can be seen that the imp. CKF almost has a larger sensitivities to the parameter uncertainties than the DCKF. The mean cost functions of three filters with logarithmic scales are shown in Fig. 9. Here, for the different cost functions of the three filters, the cost function of the perfect CKF is Jc = T r(Pk+ ), and the cost function of the imp. CKF and the DCKF is computed by using Eq. (41). It can be seen that the mean cost function of the perfect CKF is the smallest; the mean cost function of the DCKF takes second place, and the mean cost function of the imp. CKF is the largest one. So, the DCKF reduces the sensitivities of the state estimate errors respect to the parameter uncertainties, and minimizes the corresponding cost function compared with the imp. CKF. Fig. 10 shows that the RMSEs of the three filter. The RMSE for the DCKF is larger than the perfect CKF, and smaller than the imp. CKF, when the process model has the uncertain parameters. These results demonstrate that the DCKF can effectively reduce the estimation errors in the presence of the uncertain parameters. A filter consistency test based on the normalized mean error (NME) test described in reference [23] is introduced to highlight the effectiveness of the covariance estimates in acPlease cite this article as: T.-S. Lou et al., Desensitized filter with uncertain parameters, Journal of the Franklin https://doi.org/10.1016/j.jfranklin.2017.09.004
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Fig. 10. RMSE.
counting for errors in the state estimates. The results of the NME consistency test are shown in Fig. 11. It can be seen that all the state test statistics of the proposed DCKF are well below the required threshold, and this shows that the DCKF covariance estimate accurately predicts the state estimate errors. But, the results of the imp. CKF indicate that it has a poor consistency in describing the states. That is to say, the DCKF consistently provides excellent state estimates when the process model parameter has uncertainties. 5. Conclusion To overcome the effect of uncertain parameters in estimating states of nonlinear system models, we introduce the desensitized optimal control methodology to the recursive Bayesian filter, and propose the robust desensitized cubature Kalman filtering (DCKF) for nonlinear systems with uncertain parameters. The definition of the state estimate error sensitivity to the uncertain parameter is introduced to the recursive Bayesian filter. The desensitized cost function of the DCKF is designed by penalizing the posterior covariance trace by a sensitivityweighting sum of the posteriori sensitivities, and the gain of the DCKF is obtained by minimizing the desensitized cost function to amend the state estimation. In our presented DCKF, the sensitivity propagation of the state estimate errors is described, and a Lyapunov-like linear equation, which is different from the equation of Shen and Karlgaard, is solved to obtain the sensitivity of the root square matrix. Two numerical simulations are computed to verify the effectiveness of the proposed DCKF. Please cite this article as: T.-S. Lou et al., Desensitized filter with uncertain parameters, Journal of the Franklin https://doi.org/10.1016/j.jfranklin.2017.09.004
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Fig. 11. NME consistency test.
Acknowledgments This work is supported by the National Science Foundation of China (Grant Nos. 61473016, 61603346), the Fundamental Research Programs for the Provincial Universities (Grant No. 15KYYWF01), the Key research projects of Henan higher education institutions (Grant No. 18A413003) and the Doctor Startup Funds Foundation of Zhengzhou University of Light Industry (Grant No. 2015BSJJ026). The authors fully appreciate the financial supports. References [1] T. Lou, L. Zhao, Robust Mars atmospheric entry integrated navigation based on parameter sensitivity, Acta Astron. 192 (2016) 60–70. [2] A. Romanenko, L.O. Santos, P.A. Afonso, Unscented Kalman filtering of a simulated pH system, Ind. Eng. Chem. Res. 43 (2004) 7531–7538. [3] C. Wu, C. Han, Second-order divided difference filter with application to ballistic target tracking, in: 7th World Congress on Intelligent Control and Automation, IEEE, Chongqing, China, 2008, pp. 6342–6347. [4] J. Read, K. Achutegui, J.I.N.M.I. Guez, A distributed particle filter for nonlinear tracking in wireless sensor networks, Signal Process. 98 (2014) 121–134. [5] A.H. Jazwinski, Stochastic Processes and Filtering Theory, vol. 63, Academic Press, New York, 1970. [6] F.B. Hmida, K. Khmiri, J. Ragot, M. Gossa, Three-stage Kalman filter for state and fault estimation of linear stochastic systems with unknown inputs, J. Franklin Inst. 349 (2012) 2369–2388.
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[7] W. Liu, H. He, F. Sun, Vehicle state estimation based on minimum model error criterion combining with extended Kalman filter, J. Franklin Inst. 353 (2016) 834–856. [8] N.J. Gordon, D.J. Salmond, A.F.M. Smith, A novel approach TO nonlinear/non-Gaussian Bayesian state estimation, IEE Proc. F Radar Signal Process. 140 (1993) 107–113. [9] F. Daum, Nonlinear filters: beyond the Kalman filter, IEEE Aerosp. Electron. Syst. Mag. 20 (2005) 57–69. [10] I. Arasaratnam, S. Haykin, Cubature Kalman filters, IEEE Trans. Autom. Control 54 (2009) 1254–1269. [11] B. Jia, M. Xin, Y. Cheng, High-degree cubature Kalman filter, Automatica 49 (2013) 510–518. [12] X. Tang, J. Wei, K. Chen, Square-root adaptive cubature Kalman filter with application to spacecraft attitude estimation, in: 15th International Conference on Information Fusion, IEEE, Singapore, 2012, pp. 1406–1412. [13] J. Zarei, E. Shokri, Nonlinear and constrained state estimation based on the cubature Kalman filter, Ind. Eng. Chem. Res. 53 (2014) 3938–3949. [14] C. Fernandez-Prades, J. Vil-Valls, Bayesian nonlinear filtering using quadrature and cubature rules applied to sensor data fusion for positioning, in: IEEE International Conference on Communications, IEEE, Cape Town, South Africa, 2010, pp. 1–5. [15] I.Y. Song, V. Shin, S. Lee, W. Choi, Estimation fusion of nonlinear cost functions with application to multisensory Kalman filtering, J. Franklin Inst. 351 (2014) 4672–4687. [16] A. Gelb, Applied Optimal Estimation, MIT Press, MA, 1974. [17] C.D. Karlgaard, H.J. Shen, Desensitized Kalman filtering, IET Radar Sonar Navig. 7 (2013) 2–9. [18] C.D. Karlgaard, H.J. Shen, Robust state estimation using desensitized divided difference filter, ISA Trans. 52 (2013) 629–637. [19] H. Shen, C.D. Karlgaard, Sensitivity reduction of unscented Kalman filter about parameter uncertainties, IET Radar Sonar Navig 9 (2015) 374–383. [20] J.L. Crassidis, J.L. Junkins, Optimal Estimation of Dynamic Systems, second ed., Chapman & Hall/CRC Press, Boca Raton, FL, 2012. [21] J.H. Hodges, Some matrix equations over a finite field, Ann. Mat. Pura Appl. 44 (1957) 245–250. [22] D. Simon, Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches, Wiley-Interscience, New Jersey, 2006. [23] R. Madankan, P. Singla, T. Singh, P.D. Scott, Polynomial-chaos-based Bayesian approach for state and parameter estimations, J. Guidance Control Dyn. 34 (2013) 1–17.
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Journal of the Franklin Institute 000 (2017) 1–16 www.elsevier.com/locate/jfranklin
Desensitized cubature Kalman filter with uncertain parameters Tai-Shan Lou a, Lei Wang b,∗, Housheng Su c, Mao-Wen Nie d, Ning Yang a, Yanfeng Wang a a School
of Electrical and Information Engineering, Zhengzhou University of Light Industry, Zhengzhou 45002, China b School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China c School of Automation, Image Processing and Intelligent Control Key Laboratory of Education Ministry of China, Huazhong University of Science and Technology, Wuhan 430074, China d Institute of High Performance Computing, A∗ STAR, Singapore, 138632, Singapore Received 25 October 2016; received in revised form 6 August 2017; accepted 1 September 2017 Available online xxx
Abstract In this study, a robust desensitized cubature Kalman filtering (DCKF) is proposed for nonlinear systems with uncertain parameters. Unlike the cubature Kalman filtering, the desensitized cost function is introduced by penalizing the posterior covariance trace with a weighted sum of the posteriori sensitivities. The sensitivity of the root square matrix is obtained by solving a Lyapunov-like linear equation, and the sensitivity propagation of the state estimate errors is presented. The effectiveness of the proposed DCKF is demonstrated by two numerical examples in which models with uncertain parameters are considered. © 2017 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction Nonlinear state estimation plays an important role in a wide variety of applications, including estimation in navigation of aerospace vehicles [1], state of the chemical plant control [2], position and velocity of target tracking [3], and fusion of multi-sensor data [4]. A great deal of estimation methods such as extended Kalman filter (EKF), unscented Kalman filter (UKF), ∗
Corresponding author. E-mail addresses: [email protected], [email protected], [email protected] (L. Wang).
https://doi.org/10.1016/j.jfranklin.2017.09.004 0016-0032/© 2017 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
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divided difference filter (DDF), and particle filter have been proposed to solve different practical problems. However, the existing nonlinear state estimation methods still suffer from high computational cost. For instance, the EKF is computationally intensive due to unstable calculation of Jacobians or Hessians [5–7]; the UKF and DDF may not converge or converge with high computational cost [8]; the particle filter suffers from expensive computational load [9]. Recently, a nonlinear filter, called cubature Kalman filter (CKF), for the Gaussian systems has been proposed based on the Bayesian framework [10,11]. The core idea of the CKF is using the cubature role to approximate the multidimensional integrals with 2n deterministic cubature points, where n is the state vector dimension. In the CKF algorithm, the mean and the covariance of the state and the measurement vectors are calculated by propagating the cubature points through the nonlinear function. Compared with the EKF and the UKF, the CKF is usually applied in high-dimensional systems since it can produce higher accuracy and take less computational cost [10]. Subsequently, the CKF has been used in many applications including navigation [1], attitude estimation [12], chemical processing [13], and sensor data fusion [14,15], etc. As a matter of fact, all above-mentioned filters assume that the parameters of the dynamic model are accurately known as a prior and the noises of the dynamic model are white with normal probability distributions [5]. It has been shown that the performances of these filters is highly sensitive to the uncertainties of the dynamic model, and deteriorate appreciably [16]. To solve this issue, Karlgaard and Shen extended a desensitized optimal control methodology [17], and proposed a robust filter to reduce the performance sensitivity of the filters with respect to uncertain model parameters. The desensitized Kalman filter is designed by introducing a weighted norm of the state error sensitivities to the cost function, and minimizing this cost function to obtain the desensitized state estimates [17]. The desensitized divided difference filtering [18] and the desensitized unscented Kalman filtering (DUKF) [19] were presented. For the DUKF, there is no continuous propagation for the sensitivities of the sigma points between the iterations, because a new set of sigma points is always resampled at the next iteration. Shen and Karlgaard skillfully designed a unique way to propagate the sensitivities of the state estimate errors and the priori/posteriori covariance matrices for the DUKF [19]. But for the CKF, the desensitized optimal control methodology is not introduced, and the sensitivity of the root square matrix in filtering should be redesigned to avoid the calculations of the vectorized formula and Kronecker product. This paper proposes a robust cubature Kalman filter for dynamic systems with uncertain parameters. A desensitized cost function of the robust desensitized cubature Kalman filtering (DCKF) is designed by penalizing the posteriori covariance trace with a sensitivity-weighting sum of the posteriori sensitivities. Then, a gain matrix of the DCKF is obtained by minimizing the new cost function to amend the state estimation. The rest of this paper is organized as follows: Section 2 briefly introduces the recursive Bayesian framework and the CKF algorithm. Section 3 presents the propagation of the sensitivities and the DCKF is proposed naturally. Two numerical simulations about a vertically falling body model and a hovering helicopter model (HHM) are analyzed in Section 4. Conclusions are drawn in the end. 2. Cubature Kalman filter Consider a discrete nonlinear system model with additive noises, which is given by xk = f (xk−1 , c, uk−1 ) + wk−1 Please cite this article as: T.-S. Lou et al., Desensitized filter with uncertain parameters, Journal of the Franklin https://doi.org/10.1016/j.jfranklin.2017.09.004
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zk = h(xk , c, uk ) + vk
(2)
where xk ∈ Rn is the state vector at time step k, and zk ∈ Rm is the measurement vector at time step k, vector-valued functions f and h describe the nonlinear dynamic equation and the measurement equation, respectively, c ∈ R is referred to as the uncertain parameter vector, and uk ∈ Rn is the known control input vector. wk and vk are independent zero-mean Gaussian noise processes satisfying E wk wTj = Qk δi j , E vk vTj = Rk δi j , E wk vTj = 0 (3) where δ kj is the Kronecker delta function, Qk ∈ Rn×n is a positive semi-definite matrix, and Rk ∈ Rm×m is a positive definite matrix. 2.1. Recursive Bayesian filter under Gaussian domain It is easy to verify that under the Gaussian domain, the predictive density p(xk |Zk−1 ), the filter likelihood density p(zk |Zk .) and the a posterior density p(xk |Zk .) are all Gaussian, where Zk denotes the historic input measurement pairs {ui , zi }ki=1 up to time tk . Assume that + + the posterior density p(xk−1 |Zk−1 ) is N (xˆ k−1 , Pk−1 ) at time tk−1 . Then, some prerequisite distributions in filter are respectively given by p(xk |Zk−1 ) = N xˆ k− , Pk− p(zk |Zk−1 ) = N zˆ k− , Pz−z,k (4) p(xk |Zk ) = N xˆ k+ , Pk+ where N(·,·) is the Gaussian density symbol, the superscripts “− ” denotes a priori and “+ ”denotes a posteriori. xˆ k− , xˆ k+ , and zˆ k− are the state estimates and the measurement estimate, respectively; Pk− , Pk+ , and Pz−z,k are the corresponding error covariance matrices. Based on the above discussion, the functional recursion of the Bayesian filter reduces to an algebraic recursion, i.e. only the means and the covariance in Eq. (4), need to be updated at each time step. The recursive Bayesian filter under the Gaussian approximation is then summarized as follows [10]: Time update + − + dxk−1 xˆ k = f (xk−1 , c¯ , uk−1 )N xˆ k−1 , Pk−1 (5) Rn
Pk− =
Rn
+ + dxk−1 − xˆ k− xˆ k−T + Qk−1 f (xk−1 , c¯ , uk−1 )f T (xk−1 , c¯ , uk−1 )N xˆ k−1 , Pk−1
(6)
where x˜ k− = xˆ k− − xk is the priori estimation error, and c¯ is the reference value of the parameter vector c. Measurement update zˆ k− = h (xk , c¯ , uk )N xˆ k− , Pk− dxk (7) Rn
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− Pzz, k = − Pxz,k =
Rn
Rn
h (xk , c¯ , uk )hT (xk , c¯ , uk )N xˆ k− , Pk− dxk−1 − zˆ k− zˆ k−T + Rk
(8)
xk hT (xk , c, uk )N (xˆ k− , Pk− )dxk−1 − xˆ k− zˆ k−T
(9)
xˆ k+ = xˆ k− + Kk [zk − zˆ k− ]
(10)
− −T Pk+ = Pk− − Pxz,k KkT − Kk Pxz,k + Kk Pz−z,k KkT
(11)
−1
− Kk = Pxz,k (Pz−z,k )
(12)
where x˜ k+ = xˆ k+ − xk is the posteriori estimation error. Note that the Kalman optimal gain Kk in Eq. (12) can be solved by minimizing the cost function J = T r(Pk+ ), where “Tr” denotes the trace of the matrix. The detailed proof is omitted here since it can easily deduced by differentiating the covariance matrices J with respect to Kk [20]. 2.2. Cubature Kalman filter algorithm The CKF is a type of the Bayesian filter under the Gaussian assumption, i.e. the dynamic noises and the measurement noises are assumed to be zero-mean Gaussian noises. Then, the CKF approximates the mean and the covariance of a random variable propagated under a nonlinear function by the cubature rule. Under the Gaussian assumption, the functional recursion of the Bayesian filter reduces to an algebraic recursion, in which the multidimensional integrals can be approximated by the cubature rule [10]. The cubature rule approximates the n-dimensional Gaussian weighted integral via the mean μ and the covariance P, which can be formulated as
√ 1 f (x)N (μ, P)dx ≈ f μ + P ξi 2n i=1 Rn 2n
(13)
√ √ √ where P is a square-root factor of the covariance P satisfying P= P ( P )T ; ξ√i is the ith element of 2n cubature points, and its value comes from the set n{e1 , e2 , . . . , en , −e1 , −e2 , . . . , −en}, where ei (i = 1, 2, . . . , n) is the unit vector, in which the ith element is one and others are zeros, i.e. ei =[0, . . . ,0,1,0, . . . ,0]T . + + Based on the state estimate xˆ k−1 and the covariance Pk−1 at time tk−1 , the CKF is given by
+ + χ+ = Pk−1 ξ j + xˆ k−1 , j = 1, 2, . . . , 2n (14) j,k−1 + , c¯ , u χ , j = 1, 2, . . . , 2n χ ∗− = f k−1 j,k−1 j,k
(15)
1 ∗− χ 2n j=1 j,k 2n
xˆ k− =
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(16)
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1 ∗− ∗−T χ χ − xˆ k− xˆ k−T + Qk−1 2n j=1 j,k j,k
= Pk− ξ j + xˆ k− , j = 1, 2, . . . , 2n 2n
Pk− =
(17)
χ− j,k
(18)
¯ , uk ) Z−j,k = h(χ − j,k , c
(19)
1 − Z 2n j=1 j,k 2n
zˆ k− =
(20)
Pz−z,k =
1 − −T Z Z − zˆ k− zˆ k−T + Rk 2n j=1 j,k j,k
(21)
− Pxz,k =
1 − −T χ Z − xˆ k− zˆ k−T 2n j=1 j,k j,k
(22)
2n
2n
xˆ k+ = xˆ k− + Kk (zk − zˆ k− )
(23)
− −T Pk+ = Pk− − Pxz,k KkT − Kk Pxz,k + Kk Pz−z,k KkT
(24)
− Kk = Pxz,k (Pz−z,k )−1
(25)
3. Desensitized cubature Kalman filter In this section, we present a robust DCKF based on the sensitivity propagation equation and the CKF method. Section 3.1 gives the sensitivity propagation equation for the recursive Bayesian filter, and the sensitivity propagation of cubature points in the CKF is proposed in Section 3.2. 3.1. Sensitivity propagation equation for the recursive Bayesian filter Under the optimality conditions of the recursive Bayesian filter such as no model and parameter uncertainties, the state estimate are unbiased. That is to say, the estimation errors satisfy E x˜ k− = 0, E x˜ k+ = 0 (26) When the optimality conditions do not hold true, the state estimates may be biased and even divergence. So, the sensitivity of the state estimation error to each parameter, ci , can be defined as [17] ∂ x˜ k− ∂ xˆ − = k (27) ∂ci ∂ci ∂ x˜ + ∂ xˆ + + si,k = k = k (28) ∂ci ∂ci where ci denotes the ith component of the parameter vector c. Note that the sensitivity of the true state is ∂x/∂ci = 0 in Eqs. (27) and (28) since the true state does not vary with the assumed value of the parameter ci . − si,k =
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Then, the propagation equations of the recursive Bayesian filter are defined as ∂f (xk−1 , c, uk−1 ) ∂f (xk−1 , c, uk−1 ) + − + = dxk−1 si,k =E N xˆ k−1 , Pk−1 ∂ci ∂ci Rn + − si,k = si,k − Kk γi,k
(29) (30)
where the sensitivity of the nonlinear measurement function is defined by ∂h(xk , c, uk ) ∂h(xk , c, uk ) − − = γi,k = E N xˆ k , Pk dxk ∂ci ∂ci Rn
(31)
Note that the sensitivity of the gain matrix should be satisfied with ∂K/∂ci = 0 in Eq. (30). One key reason is that if ∂ K/∂ ci = 0, then the optimal gain is a function of the residual violating the basis for the Lyapunov-like linear update equation given in Eq. (11) [17]. 3.2. Sensitivity propagation of cubature points According to the partial derivative of the cubature point generated equation, the propagation of the sensitivities is determined bases on the sensitivity of cubature points, such as Eqs. (14) and (18). Then, the propagation equations of the sensitivities can be obtained by taking partial derivative of the propagation equations of the CKF, such as Eqs. (15)–(17) and (19)– (24). The sensitivity propagation algorithm of the cubature points is summarized as following: First, compute the sensitivities of step k − 1 cubature points (i = 1, 2, . . . , , j = 1, 2, . . . , 2n)
+ ∂ Pk−1 ∂χ + j,k−1 + = ξ j + si,k−1 , (32) ∂ci ∂ci and update the sensitivity cubature points + , c¯ , u + ¯ χ c ∂f χ + , , u ∂f k−1 k−1 ∂χ ∗− ∂χ j,k−1 j,k−1 j,k−1 j,k = + ∂ci ∂ci ∂ci ∂χ +
(33)
j,k−1
Second, evaluate the sensitivities of the prior state estimate and the prior covariance matrix ∗−
∂ xˆ k− 1 ∂χ j,k = ∂ci 2n j=1 ∂ci ⎧ ∗− T ⎫ ∗− 2n ⎨ ⎬ − ∂χ ∂χ j,k ∂Pk 1 j,k ∗−T − −T −T = χ j,k + χ ∗− − si,k xˆ k − xˆ k− si,k j,k ⎭ ∂ci 2n j=1 ⎩ ∂ci ∂ci 2n
− si,k =
(34)
(35)
Third, compute the sensitivities of the redrawn cubature points and the predicted measurement cubature points
∂ Pk− ∂χ − j,k − = ξi + si,k , (36) ∂ci ∂ci ¯ , uk ) ∂χ − ¯ , uk ) ∂Z−j,k ∂h(χ − ∂h(χ − j,k , c j,k j,k , c = + , (37) − ∂ci ∂ci ∂ci ∂χ j,k Please cite this article as: T.-S. Lou et al., Desensitized filter with uncertain parameters, Journal of the Franklin https://doi.org/10.1016/j.jfranklin.2017.09.004
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and evaluate the sensitivities of the predicted measurement −
∂ zˆ k− 1 ∂Z j,k = ∂ci 2n j=1 ∂ci 2n
γi,k =
(38)
Fourth, evaluate the innovation covariance matrix and the cross-covariance matrix ⎧ − T ⎫ − 2n ∂Z j,k ⎬ ∂Pz−z,k 1 ⎨ ∂Z j,k −T T = Z j,k + Z−j,k − γi,k zˆ k−T − zˆ k− γi,k ⎭ ∂ci 2n j=1 ⎩ ∂ci ∂ci ⎧ − T ⎫ − 2n − ∂Z j,k ⎬ ∂Pxz,k 1 ⎨ ∂χ j,k −T − −T T = Z j,k + χ − − si,k zˆ k − xˆ k− γi,k j,k ⎩ ∂ci ⎭ ∂ci 2n ∂ci
(39)
(40)
j=1
Lastly, evaluate the sensitivities of the posterior state estimate and the posterior covariance matrix + − si,k = si,k − Kk γi,k
(41)
− −T ∂Pxz,k ∂Pxz,k ∂Pz−z,k T ∂Pk+ ∂Pk− = − KkT − Kk + Kk Kk (42) ∂ci ∂ci ∂ci ∂ci ∂ci √ The sensitivity of the root square matrix P in Eqs. (32) and (36) must be computed when the sensitivities of the cubature points √ √ and redrawn cubature points are calculated. By taking partial derivative of equation P= P ( P )T , which is different from the vectorized formula in Karlgaard and Shen [17], we have √ √ ∂P ∂ P √ T √ ∂ P T = ( P ) + P( ) (43) ∂ci ∂ci ∂ci √ The solution ∂ P/∂ci in Eq. (43) can be obtained as [21] −1 √ T ∂ P ∂P −1 1 T = −
T (44) ∂ci 2 ∂ci
where ∈ Rn×n is a √ skew symmetric matrix such that T = − , and are non-singular matrices satisfying P = I . 3.3. Desensitized cubature Kalman filter algorithm In addition to propagate the sensitivity of cubature points in Section 3.2 and compute the gain matrix different from the CKF, other steps of the proposed DCKF are the likely same as the CKF. So, only the gain matrix is derived in this section. With the sensitivities of the posterior state estimate in Section 3.2, a desensitized cost function of the DCKF consisting of the posterior covariance and a weighted norm of the posterior sensitivity is penalized by + a sensitivity-weighting sum of the sensitivities si,k Jd = T r(Pk+ ) +
+T + si,k Wi,k si,k
(45)
i=1
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Table 1 DCKF algorithm. Step 0: Set k = 0, and initialize the state vector, xˆ 0 , the auxiliary matrix, P0 , and the sensitivity parameters, s0 and ∂P0 /∂ci , i = 1, 2, . . . , ; ∗− Step 1: Evaluate the propagated cubature points, χ ∗− j,k , using (14), and the sensitivities, ∂χ j,k /∂ci , by Eqs. (44) and (32); − Step 2: Estimate the predicted state, xˆ k− , by Eq. (15), and the prior sensitivity, si,k , by Eq. (33); Step 3: Evaluate the propagated cubature points, χ ∗− , by Eq. (16) , and the sensitivities, ∂χ ∗− j,k j,k /∂ci , by Eq. (34); − Step 4: Estimate the predicted covariance matrix, Pk , by Eq. (17), and the sensitivity, ∂Pk− /∂ci , by Eq. (35); + Step 5: Evaluate the redrawn cubature points, χ − j,k , by Eq. (18), and the sensitivities, ∂χ j,k−1 /∂ci , by Eqs. (44) and (36); Step 6: Evaluate the propagated cubature points of measurement equation, Z−j,k , by Eq. (19), and the sensitivities, ∂Z−j,k /∂ci , by Eq. (37); Step 7: Estimate the predicted measurement, zˆ k− , by Eq. (20), and the sensitivity, γ i, k , by Eq. (38); Step 8: Estimate the innovation covariance matrix, Pz−z,k , by Eq. (21), and the sensitivity, ∂Pz−z,k /∂ci , by Eq. (39); − − Step 9: Estimate the cross-covariance matrix, Pxz,k , by Eq. (22), and the sensitivity, ∂Pxz,k /∂ci , by Eq. (40); Step 10: Obtain the gain matrix, Kk , by solving Eq. (46); + Step 11: Estimate the posterior state, xˆ k+ , by Eq. (23), and the posterior sensitivity, si,k , by Eq. (41); + Step 12: Estimate the posterior covariance matrix, Pk , by Eq. (24), and the sensitivity, ∂Pk+ /∂ci , by Eq. (42); Step 13: Set k = k + 1 and return to step 1.
where Wi,k ∈ Rn×n is a symmetric positive semi-definite weighting matrix for the ith sensitivity. Substituting Eqs. (24) and (41) into Eq. (45), taking partial derivative of Jd with respected to the gain matrix Kk , and setting the partial derivative ∂Jd /∂Kk = 0 gives the solution Kk Pz−z,k
+
T Wi,k Kk γi,k γi,k
i=1
=
− Pxz,k
+
− T Wi,k si,k γi,k
(46)
i=1
where the gain Kk is obtained by algebraically solving with the Lyapunov-like linear equation in Eq. (46). The DCKF algorithm is summarized in Table 1. 4. Numerical simulation To demonstrate the effectiveness of the proposed DCKF algorithm in the previous section, the vertical falling body model with one uncertain parameter [20,22] and the HHM with two uncertain parameters are considered [23]. The perfect CKF (perf. CKF) and the imperfect CKF (imp. CKF) are employed to compare the performance of the proposed algorithm. The “perfect” means that the true values of the parameters are all known exactly in perf. CKF; the “imperfect” means that in the imp. CKF the true values of the uncertain parameters are not known, and only the reference values, coming from the previous experience, are known. 4.1. Falling body model with one uncertain parameter In this example, we use the perf. CKF, the imp. CKF and the proposed DCKF to estimate the altitude x1 (t), the velocity x2 (t) and the constant ballistic coefficient x3 (t) of a vertically falling body with a high velocity [20,22]. A tracking radar device is used to record the range measurements between the radar and the falling body. The system equation is given by Please cite this article as: T.-S. Lou et al., Desensitized filter with uncertain parameters, Journal of the Franklin https://doi.org/10.1016/j.jfranklin.2017.09.004
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Fig. 1. Sensitivities of the imperfect CKF and the DCKF for state x1 .
Fig. 2. Sensitivities of the imperfect CKF and the DCKF for state x2 .
Fig. 3. Sensitivities of the imperfect CKF and the DCKF for state x3 .
x˙1 (t ) = x2 (t ) + w1 (t ) x˙1 (t ) = x22 (t )x3 (t ) exp{−x1 (t )/c} − g + w2 (t ) x˙3 (t ) = w3 (t )
(47)
where constant c reflects relationship between the air density and the altitude with its reference value c¯ = 2 × 104 , g = 32.2 ft/s2 is the gravitational acceleration. wi (t ) is the process Please cite this article as: T.-S. Lou et al., Desensitized filter with uncertain parameters, Journal of the Franklin https://doi.org/10.1016/j.jfranklin.2017.09.004
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Fig. 4. x1 RMSE.
Fig. 5. x2 RMSE.
Fig. 6. x3 RMSE.
noise that affects the ith equation with E [wi2 (t )] = 0 (i = 1, 2, 3). The discrete-time range measurement from the radar is given by zk =
M 2 + (x1,k − H )2 + vk
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(48) cubature Institute
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Fig. 7. True values of the HHM and estimates of the imp. CKF and the DCKF.
where M = 105 ft is the radar horizontal distance from the body’s vertical line of fall, H = 105 ft is the radar altitude above the ground level, and vk is the measurement noise assumed to be a zero-mean Gaussian noise with covariance E [vk2 ] = R = 104 ft2 . Also, assume that the constant parameter c is uncertain, and the true values of c follows a uniform distribution, c ∼ U (3/4c¯, 5/4c¯). The true state and initial estimates and the initial covariance are given as x1 (0) = 3 × 105 ft; x2 (0) = −2 × 104 ft/s; x3 (0) = 1 × 10−3
(49)
xˆ1 (0) = 3 × 105 ft; xˆ2 (0) = −2 × 104 ft/s; xˆ3 (0) = 3 × 10−5
(50)
P0+ = diag 1 × 106 ft2 , 4 × 106 ft2 /s2 , 1 × 10−4
(51)
The total simulation time is fixed at 60 s, and the fourth-order Runge–Kutta method is used to discretize the continuous state equation with a sample frequency 10 Hz. 200 Monte Carlos are run to evaluate the performance of the three filters. In the imp. CKF and the DCKF, the uncertain parameter c is set as the corresponding reference value c¯ = 2 × 104 . For the DCKF, the sensitivity-weighting matrix is set to (52) W = diag 3 × 104 , 6 × 103 , 1 × 105 The state sensitivities with respect to the uncertain parameter c with logarithmic scales are shown in Figs. 1–3. Compared to the imp. CKF, the proposed DCKF almost has smaller sensitivities to the uncertain parameter. Figs. 4–6 show that the root mean squared errors Please cite this article as: T.-S. Lou et al., Desensitized filter with uncertain parameters, Journal of the Franklin https://doi.org/10.1016/j.jfranklin.2017.09.004
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Fig. 8. Sensitivities of the imp. CKF and the DCKF.
(RMSEs) of the above three filters. When the process model is disturbed by the uncertain parameter, the RMSE for the DCKF is smaller than that of the imperfect CKF. In a word, the proposed DCKF can reduce the sensitivities of the state estimate errors respect to the uncertain parameter compared with the imperfect CKF, and has a better performance than the imperfect CKF when the process model parameter is imperfect. 4.2. Hovering helicopter model with two uncertain parameters In this example, the ⎡ c1 c2 ⎢1.26 −1.765 x˙ = ⎢ ⎣ 0 1 1 0
hovering helicopter model is given by [23] ⎤ ⎤ ⎡ −g 0 0.086 ⎢ ⎥ 0 0⎥ ⎥x − ⎢−7.408⎥Klqr x ⎣ 0 ⎦ 0 0⎦ 0 0 0
(53)
zk = xk + vk
(54)
where the state vector is x = [x1 , x2 , x3 , x4 ]T , in which x1 is the horizontal velocity, x2 is the pitch angle of the fuselage and its derivative x3 , and x4 is perturbation from a ground point reference. g is the acceleration from gravity and its value is g = 0.322, Klqr is a constant vector given by Klqr = [1.989, −0.256, −0.7589, 1]. vk is the measurement noise. The two uncertain model parameters c1 and c2 are assumed to be uniform distributions, which are Please cite this article as: T.-S. Lou et al., Desensitized filter with uncertain parameters, Journal of the Franklin https://doi.org/10.1016/j.jfranklin.2017.09.004
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Fig. 9. Mean cost function.
respectively c1 ∼ U (−0.15, −0.05) and c2 ∼ U(0.05, 0.15). The initial true state of the HHM is x0 = [0.7929, −0.0466, −0.1871, 0.5780]T
(55)
and the initial state of the filter is set as xˆ 0 = x0 , with the initial covariance Pˆ 0 = I . In the simulation, the total time interval is assumed to be 4s, and the continuous state equation has been discretized using the fourth-order Runge–Kutta method with a sample frequency 20 Hz. For performance evaluation, 200 Monte Carlo runs are performed for three filters. In the imp. CKF and the DCKF the two uncertain parameters are set as their reference values, which are c¯1 = −0.1 and c¯2 = 0.1. For the DCKF, the sensitivity-weighting matrices are set to (56) W1 = W2 = diag 3 × 10−3 , 2 × 10−3 , 1 × 10−2 , 2 × 10−2 The true state of the HHM and its estimates of the imp. CKF and the DCKF in one Monte Carlo test are shown in Fig. 7. The true values of the two parameters in this simulation case are c¯1 = −0.0729 and c¯2 = 0.1142, respectively. Fig. 8 represents the sensitivity of the state variables to the parameters, c1 , in the imp. CKF and the DCKF with logarithmic scales, because the state sensitivities to c2 are similar. From Fig. 8, it can be seen that the imp. CKF almost has a larger sensitivities to the parameter uncertainties than the DCKF. The mean cost functions of three filters with logarithmic scales are shown in Fig. 9. Here, for the different cost functions of the three filters, the cost function of the perfect CKF is Jc = T r(Pk+ ), and the cost function of the imp. CKF and the DCKF is computed by using Eq. (41). It can be seen that the mean cost function of the perfect CKF is the smallest; the mean cost function of the DCKF takes second place, and the mean cost function of the imp. CKF is the largest one. So, the DCKF reduces the sensitivities of the state estimate errors respect to the parameter uncertainties, and minimizes the corresponding cost function compared with the imp. CKF. Fig. 10 shows that the RMSEs of the three filter. The RMSE for the DCKF is larger than the perfect CKF, and smaller than the imp. CKF, when the process model has the uncertain parameters. These results demonstrate that the DCKF can effectively reduce the estimation errors in the presence of the uncertain parameters. A filter consistency test based on the normalized mean error (NME) test described in reference [23] is introduced to highlight the effectiveness of the covariance estimates in acPlease cite this article as: T.-S. Lou et al., Desensitized filter with uncertain parameters, Journal of the Franklin https://doi.org/10.1016/j.jfranklin.2017.09.004
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Fig. 10. RMSE.
counting for errors in the state estimates. The results of the NME consistency test are shown in Fig. 11. It can be seen that all the state test statistics of the proposed DCKF are well below the required threshold, and this shows that the DCKF covariance estimate accurately predicts the state estimate errors. But, the results of the imp. CKF indicate that it has a poor consistency in describing the states. That is to say, the DCKF consistently provides excellent state estimates when the process model parameter has uncertainties. 5. Conclusion To overcome the effect of uncertain parameters in estimating states of nonlinear system models, we introduce the desensitized optimal control methodology to the recursive Bayesian filter, and propose the robust desensitized cubature Kalman filtering (DCKF) for nonlinear systems with uncertain parameters. The definition of the state estimate error sensitivity to the uncertain parameter is introduced to the recursive Bayesian filter. The desensitized cost function of the DCKF is designed by penalizing the posterior covariance trace by a sensitivityweighting sum of the posteriori sensitivities, and the gain of the DCKF is obtained by minimizing the desensitized cost function to amend the state estimation. In our presented DCKF, the sensitivity propagation of the state estimate errors is described, and a Lyapunov-like linear equation, which is different from the equation of Shen and Karlgaard, is solved to obtain the sensitivity of the root square matrix. Two numerical simulations are computed to verify the effectiveness of the proposed DCKF. Please cite this article as: T.-S. Lou et al., Desensitized filter with uncertain parameters, Journal of the Franklin https://doi.org/10.1016/j.jfranklin.2017.09.004
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Fig. 11. NME consistency test.
Acknowledgments This work is supported by the National Science Foundation of China (Grant Nos. 61473016, 61603346), the Fundamental Research Programs for the Provincial Universities (Grant No. 15KYYWF01), the Key research projects of Henan higher education institutions (Grant No. 18A413003) and the Doctor Startup Funds Foundation of Zhengzhou University of Light Industry (Grant No. 2015BSJJ026). The authors fully appreciate the financial supports. References [1] T. Lou, L. Zhao, Robust Mars atmospheric entry integrated navigation based on parameter sensitivity, Acta Astron. 192 (2016) 60–70. [2] A. Romanenko, L.O. Santos, P.A. Afonso, Unscented Kalman filtering of a simulated pH system, Ind. Eng. Chem. Res. 43 (2004) 7531–7538. [3] C. Wu, C. Han, Second-order divided difference filter with application to ballistic target tracking, in: 7th World Congress on Intelligent Control and Automation, IEEE, Chongqing, China, 2008, pp. 6342–6347. [4] J. Read, K. Achutegui, J.I.N.M.I. Guez, A distributed particle filter for nonlinear tracking in wireless sensor networks, Signal Process. 98 (2014) 121–134. [5] A.H. Jazwinski, Stochastic Processes and Filtering Theory, vol. 63, Academic Press, New York, 1970. [6] F.B. Hmida, K. Khmiri, J. Ragot, M. Gossa, Three-stage Kalman filter for state and fault estimation of linear stochastic systems with unknown inputs, J. Franklin Inst. 349 (2012) 2369–2388.
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