Study on initial value problem for fractional-order cubature Kalman filters of nonlinear continuous-time fractional-order systems

To realize the state estimation of nonlinear continuous-time fractional-order systems, two types of fractional-order cubature Kalman filters are designed to solve problem on the initial value influence. For the first type of cubature Kalman filter (CKF), the initial value of the estimated system is also regarded as the augmented state, and the augmented state equation is constructed to obtain the CKF based on Grünwald–Letnikov difference. For the second type of CKF, the fractional-order hybrid extended-cubature Kalman filter is proposed to weaken the influence of initial value by the first-order Taylor expansion and the third-order spherical-radial rule. These two methods can reduce the influence of initial value on the state estimation effectively. Finally, the effectiveness of the proposed CKFs is verified by two simulation examples.


Introduction
In recent decades, fractional-order systems (FOSs) has been studied by many scholars, because fractionalorder calculus can more accurately describe the phenomena in real-world systems [1][2][3]. The fractionalorder calculus is applied to deal with the fractionalorder integral and derivative, and it is widely used in the network system [4], the heat conduction [5], the material science [6] and so on. The fractionalorder controller is a common application of fractionalorder calculus [7]. Due to the simple structure of fractional-order PID controller, the fractional-order PID controller is used in most industrial control systems [8,9]. With the introduction of fractional-order operators, other types of fractional-order controllers appear, such as the fractional-order sliding mode controller [10], the fractional-order back-stepping controller [11], the fractional-order model predictive controller [12] and the fractional-order extremum seeking controller [13]. In practical control systems, the measurement signal inevitably contains certain noise disturbances, and some state information may not be obtained directly. Then robust state observers for FOSs need to be designed.
As one kind of robust state observers [14], the Bayesian filtering can achieve the state estimation of nonlinear systems. The key of solving the nonlinear filtering problem is to obtain the posterior probability distribution of the state information. Thus, the prob-lem of nonlinear filtering can be solved in the framework of Bayesian filtering [15]. As a special case of Bayesian filter, Kalman filter (KF) has become an increasing common tool to estimate the state information of integer-order systems (IOSs). The extended Kalman filter (EKF) is a very effective method to deal with nonlinear filtering problem based on the first-order Taylor expansion [16]. EKF has been widely used in the fields of the state-of-charge estimation of lithium-ion battery [17], the multi-sensor tracking of maneuvering targets [18] and the current transformer saturation [19]. However, the state estimation accuracy of EKF linearization method is not ideal, sometimes. If some nonlinear functions are not differentiable, EKF cannot effectively estimate the state information, which is also the limitation of EKF [20]. Hence, other types of KFs have been designed to deal with nondifferentiable nonlinear functions, such as unscented Kalman filter (UKF) [21][22][23] and cubature Kalman filter (CKF) [24][25][26]. The UKF uses the unscented transformation to cope with nonlinear functions in FOSs, then more accurate state information is obtained by comparing with EKF [27].
Meanwhile, fractional-order Kalman filters (FOKFs) are still appropriate to estimate state information for linear and nonlinear FOSs. The FOKFs can obtain more satisfactory state estimation than that of integer-order KFs. The fractional-order EKFs were proposed for linear FOSs with white Gaussian noises and colored noises in [28,29], respectively. In [30], KFs for linear and nonlinear discrete-time FOSs were extended to estimate the parameters or order of FOSs. Also, linear and nonlinear discrete-time FOSs were mentioned in [30]. In [31], the improved fractional-order UKF for nonlinear FOSs was designed to estimate more accurate state information. Besides, CKF can also estimate effectively state information for nonlinear FOSs. In [32], the CKF method was applied to obtain the higher estimation accuracy compared with EKF method for nonlinear continuous-time FOSs containing white Gaussian noises. Meanwhile, hybrid FOKFs for nonlinear FOSs were also offered to enhance the estimation accuracy in [33,34].
For a linear FOS, the initial value performs a significant impact on the state estimation and fractionalorder identification if the fractional-order of the system is relatively small in the interval (0, 1), then an initial value compensation (IVC) is achieved by means of an augmented vector. The EKF with IVC for lin-ear continuous-time FOSs with an unknown fractionalorder was proposed in [35]. In this paper, the CKF for nonlinear FOSs with the IVC is investigated to obtain more effective state estimation. This paper introduces the Grünwald-Letnikov (G-L) difference method into two types of fractional-order cubature Kalman filters (FOCKFs) to obtain the effective state estimation of a nonlinear FOS with IVC.
The main contributions of this paper are generalized as follows: (1) The state estimation problem of nonlinear FOSs is investigated. (2) For a relatively small fractional-order in (0, 1), the augmented vector method is used to achieve the IVC. (3) The influence of initial value on state estimation is weakened by the model transformation method. (4) The FOCKF with the IVC and the fractional-order hybrid extended-cubature Kalman filter (HECKF) for nonlinear continuous-time FOSs is proposed to achieve the higher accuracy state estimation to compare with the CKF without the IVC in [32].
The remaining of this paper is arranged as follows. In Sect. 2, we introduce the definitions of fractionalorder derivatives, the third-order spherical-radial rule and cubature points. We design the discretized equation of a nonlinear FOS, the FOCKF and HECKF to solve initial value problems in Sect. 3, respectively. In Sect. 4, the simulation examples are given to testify the validity of the proposed FOCKF and HECKF with the IVC. Section 5 summarizes this paper.

Definitions of fractional-order derivatives
In the subsection, some definitions for fractional-order derivatives are introduced to describe dynamic characteristics of nonlinear FOSs. Considering the consistency of initial conditions of FOS, the Caputo definition has an increasingly wide application in real-world systems. Therefore, the establishment of nonlinear FOSs is achieved based on the Caputo definition and the IVC is considered, and the Riemann-Liouville (R-L) definition is also used in this study.

Definition 1
The R-L definition [36] with α-order derivative for ε(t) is defined as represents α-order derivative based on R-L definition, α ∈ R + is the fractional-order and satisfies n − 1 ≤ α < n, n ∈ Z + , R + and Z + are the set of positive numbers and the set of positive integer numbers, respectively, and Γ (·) is the Gamma function.
It is obvious that the main difference between Definitions 1 and 2 is the order of differential operation and integral operation. In addition, the relationship between Definitions 1 and 2 is described by [

Definition 3
The G-L difference [38] for α-order is defined by where Δ α is the α-order difference operation, T is the sampling period, and the coefficient b α m is represented as follows Since ε(r − m) = 0 for r > m, the R-L definition is approximately equivalent to G-L difference at t = r T , then For α ∈ (0, 1), we take n = 1 in (1), then it follows that where To describe the impact of the initial value ε(0), we define ρ(r ) = |A(r )|. Figure 1 shows the relationship between ρ(r ) and the iteration r .  Fig. 1, it is seen that ρ(r ) decreases gradually with the increase of r . If the fractional-order α becomes closer to 1, ρ(r ) decays faster with an increasing of r . An decreasing fractional-order has a larger influence on the initial value. Therefore, the IVC can estimate more accurate state information for the FOS with small fractional-order in (0, 1).

Third-order spherical-radial rule and construction of cubature points
The EKF algorithm uses Taylor series to expand the nonlinear function and retains the first-order term to realize the linearization of a nonlinear function. Other high-order terms are ignored, then the estimation accuracy becomes greatly reduced. Besides, the CKF algorithm is available if the nonlinear function is nondifferentiable. Due to the simple structure of EKF algorithm, it is suitable for systems with differentiable nonlinear functions. The CKF algorithm is performed based on the third-order spherical-radial rule via the cubature points to deal with the nonlinear function with a higher precision and suitable for the nondifferentiable nonlinear function. The vector ε ∈ R n satisfies the Gaussian distribution, namely ε ∼ N (ε;ε, M), whereε represents the mathematical expectation of ε, the covariance matrix of ε is represented by M with the following transfor- The nonlinear function l(ε) satisfies the following equation The 2n cubature points are gained by performing the transformation and e i ∈ R n is the unit vector on the ith axis. Based on the third-degree spherical-radial rule in [24], we use the 2n cubature points defined in (5), and the integral R n l(ε + √ Mγ )N (γ ; 0, I )dγ in (4) is approximately described by Combining Eq. (6) with Eq. (4), the following equation holds 3 Main results 3.1 Fractional-order cubature Kalman filter with IVC by using augmented vector method Consider the following nonlinear continuous-time FOS as where ϕ(ε(t), u(t)) and y(ε(t)) represent the nonlinear functions, α ∈ (0, 1) is fractional-order, ε(t) ∈ R n , u(t) ∈ R p and η(t) ∈ R q represent the state vector, the input and the measurement output, respectively, w(t) ∈ R n and v(t) ∈ R q are process and measurement noises, and satisfy

It follows that
where The vector ε(r ) defined as the initial state ε(0) at the r th iteration satisfies where w(r ) represents the white Gaussian noise and In order to reduce the influence of initial value on a FOS, we define the augmented vector Based on Eqs. (10) and (11), the corresponding augmented equation is given as follows where the nonlinear function an identity matrix with the appropriate dimension. A CKF algorithm with the IVC is proposed to achieve more accuracy estimation effect of nonlinear FOS (8). The prediction and estimation of ζ(r ) are defined asζ (r |r , the predictionζ (r |r − 1) is obtained on the basis of difference Eq. (12) as follows The probability density p(ζ (r − 1)|λ(r − 1)) is supposed to satisfy the Gaussian distribution, that is Therefore, the predictionζ (r |r − 1) is derived bŷ [38], and the prediction error matrix M(r |r − 1) is defined as Then, the prediction error matrix P(r |r − 1) is derived as follows The probability density of ζ(r ) is supposed to satisfy the Gaussian distribution. Based on the prediction ζ (r |r − 1) and the prediction error matrix M(r |r − 1), hencê Thus, the Kalman gain matrix K (r ) is represented as follows where The updating formulas forζ (r |r ) and M(r |r ) are shown, respectively, as followŝ Based on the approximate method for numerical integration (7), the cubature points are defined by where γ (i) is given in Eq. (5). Therefore, the prediction and the prediction error matrix are presented as follows, respectively.
Remark 1 Because the storage capacity is limited, we select a suitable truncation L in the practical engineering. Therefore,ζ (r |r − 1) and M(r |r − 1) with a truncation L are modified bŷ and M(r |r − 1)

Hybrid extended-cubature Kalman filter to weaken impact of initial value
The advantages of EKF algorithm and CKF algorithm are concerned to weaken the influence of initial value in this subsection. Based on the definition of Caputo derivative, the following equation for α ∈ (0, 1) is taken as Therefore, it follows that ε (1) Using the G-L difference to discretize Eq. (25) at t = r T , the following equation is gained as where the absolute value of the coefficient for It is difficult to design fractionalorder Kalman filter since the noise term is related to historical information, hence the noise term needs to be simplified. The coefficient b 1−α m decreases rapidly with the increases of m, while b 1−α m tends to zero if m becomes very large. We retain the two terms w(r ) and w(r − 1), then Eq. (26) is approximated by The estimation of ε(r ) is denoted byε(r |r ) = E[ε(r )|λ(r )], where λ(r ) covers η(0), η(1), . . . , η(r ), u(0), u(1), . . . , u(r ), hence the first-order Taylor expansion of the nonlinear function ϕ(ε(r ), u(r )) at ε(r ) =ε(r − 1|r − 1) is given as follows where C(r − 1) = ∂ϕ(ε,u(r )) ∂ε | ε=ε(r −1|r −1) . For m ≥ 2, the first-order Taylor expansion of the nonlinear function ϕ(ε(r −m), u(r −m)) at ε(r −m) = ε(r − m|r − m) is represented by where N (r − m) = ∂ϕ(ε,u(r −m)) ∂ε | ε=ε(r −m|r −m) .
Being similar to the cubature points in (17) and (20), we define the cubature points as follows and Then, the predictionε(r |r − 1), the prediction error matrix M(r |r − 1),η(r |r − 1), P(r ) and Z (r ) are approximately represented bŷ and

Remark 2
Because the storage capacity is limited, we select a suitable truncation L in the practical engineering. Therefore,ε(r |r − 1) and M(r |r − 1) with a truncation L are modified bŷ ε(r |r − 1) The CKF and HECKF are proposed to weaken the initial value influence on state estimation for nonlinear FOS (8) compared with Algorithm 1 based on the CKF in [32].
For a large dimension of the state vector ε(r ), the augmented vector method may lead to a sharp increase in the amount of calculation, CKF algorithm with augmented vector method is not suitable for the system with a large dimension, hence, we select HECKF algorithm. However, in the case of strong nonlinearity or nondifferentiability of nonlinear function in the estimated system, HECKF algorithm cannot estimate the state information of a nonlinear FOS effectively, then CKF algorithm is selected. Therefore, we need to choose the suitable method according to the characteristics of actual systems.

Numerical examples
Example 1 is given to study the state estimation for nonlinear FOS (39) and verify the validity of the proposed Algorithm 1 based on the CKF. Meanwhile, the calculation time is compared with Algorithm 1 based on the CKF in [32]. Example 2 is used to investigate the state estimation for nonlinear FOS (40) and check the validity of the proposed Algorithm 2 based on the HECKF. Moreover, the calculation time for state estimation is compared with Algorithm 1 based on the CKF in [32].
To evaluate the estimation impact by the proposed Algorithm 1 based on CKF with the IVC, the following indicator is set by where S + 1 is the number of the sampling values of ε(r ).  Let the truncation L be L = 25, . . . , 50 with step 5, the fractional-order α is set from 0.2 to 0.8 with the step 0.1, then the estimation error E for Algorithm 1 is shown in Table 1.
Hence, the estimation error E of system (39) has a tendency to increase for the fractional-order α setting from 0.2 to 0.8 with the step 0.1, with the increase of truncation L. However, the calculation burden is increased by a larger truncation L. To balance the estimation accuracy and the calculation complexity, an appropriate truncation L is needed to be selected in a practical engineering. To verify the effectiveness of the proposed Algorithm 1 to estimate the state information of system (39), we select L = 35 and α = 0.3. Then, Fig. 2 draws the measurement η(r ).   of iteration r is relatively small, the estimation of initial value is also preformed in the dynamic adjustment. Therefore, the responses of the state estimations may fluctuate before the initial value reaches the real value. Figures 3 and 4 show that the proposed Algorithm 1 based on the CKF can more effectively estimate the To compare with the estimation effect of the CKF without the IVC in [32] and the proposed Algorithm 1 with the IVC in this paper, we set the fractional-order as 0.2, 0.3, . . . , 0.8, and the truncation as L = 35, then the error indexes under the index E are shown in Table 2, where E 1 and T 1 represent estimation error and calculation time of Algorithm 1 in [32], respectively. E 2 and T 2 are estimation error and calculation time of the proposed Algorithm 1 in this paper, respectively. The simulation software is MATLAB 2014a, hardware configuration is AMD A10-5750M APU with Radeon(tm) HD Graphics, the processing speed of CPU and the RAM are 2.50 GHz and 4.0 GB, respectively.
From Table 2, it is obvious that the state estimation based on CKF proposed by Algorithm 1 is more accurate for different fractional-orders α = 0.2, 0.3, . . . , 0.8, but a longer time is taken for calculating. Besides, the difference between E 1 and E 2 via Algorithm 1 and Algorithm 1 in [32] decreases with increasing of fractional-order α, and it indicates that the effect of state estimation based on CKF via Algorithm 1 with IVC is more obvious if the fractional-order α ∈ (0, 1) is relatively small. Thus, the CKF based on Algorithm 1 can be applied to estimate the state information for a nonlinear continuous-time FOS with a relatively small fractional-order α ∈ (0, 1), and the CKF based on Algorithm 1 in [32] is more suitable for the system with a relatively large fractional-order α ∈ (0, 1) if the calculation time of state estimation is concerned.
The sampling period of model discretization also determines the dynamic characteristics of a nonlinear fractional-order system. We set the sampling period T from 0.002 s to 0.25 s with the step 0.008 s.
Then, the curves of Algorithm 1 and Algorithm 1 in [32] under different sampling periods T are shown in Fig. 5 with L = 35.
At T = 0.002 s, the estimation error of Algorithm 1 is slightly larger than that of Algorithm 1 in [32], but the estimation errors via two algorithms are both relatively large. After T = 0.02 s, the estimation errors present by Algorithm 1 are clearly smaller than that given by Algorithm 1 in [32]. The sampling period should be selected reasonably in terms of an actual model. If the sampling period is set too large or too small, the estimation error may increase. If the estimation errors of the two algorithms are both large, it means that the corresponding sampling period T is not appropriate.
From Table 3, the index E becomes smaller with increasing of the truncation L for the fractional-order α set from 0.2 to 0.8 with the step 0.1. A small estimation error of FOS (40) with a relatively small fractionalorder α in (0, 1) is still obtained by the proposed Algorithm 2 based on the HECKF. To testify the effectiveness of the proposed Algorithm 2 for the state estimation, we select L = 35 and α = 0.3. Then Fig. 6 offers η(r ) in Example 2. Meanwhile, Figs. 7 and 8, respectively, draw the real values and the estimations of ε 1 (r ) and ε 2 (r ). The red and blue curves in Figs. 7 and 8 represent the estimation error curves of Algorithm 2 considering the influence of initial value and Algorithm 1 without the IVC in [32], respectively. It can be seen that the proposed Algorithm 2 based on the HECKF can usually obtain more accurate state estimation, compared with Algorithm 1 based on CKF in [32] from the curves of Figs. 7 and 8. It verifies that the model transformation method can weaken the influence of initial value on state estimation.
The estimation effect of Algorithm 1 in [32] and the proposed Algorithm 2 based on the HECKF in this paper is evaluated by the error index E. The fractionalorder α is set as 0.2, 0.3, . . . , 0.8, the truncation L = 35, and the error indexes under the index E are shown in Table 4, where E 3 and T 3 represent the estimation error and the calculation time of Algorithm 1 in [32], respectively. The E 4 and T 4 are estimation error and calculation time of proposed Algorithm 2 in this paper, respectively.    Table 4, it is obvious that the state estimation based on CKF by the proposed Algorithm 2 is more accurate for different fractional-orders α = 0.2, 0.3, . . . , 0.8, but it needs to take a longer time for calculation. In addition, the difference of state esti-  Fig. 9 Estimation errors of Algorithm 2 and Algorithm 1 in [32] mation errors between Algorithm 2 and Algorithm 1 in [32] gradually decreases as the fractional-order α increases, and a more accurate estimation effect of Algorithm 2 based on CKF for a relatively small fractional-order α ∈ (0, 1) is obtained to compare with Algorithm 1 in [32]. Moreover, the CKF based on Algorithm 1 in [32] is more applicable for FOS (40) with a relatively large fractional-order α ∈ (0, 1) if the calculation time of state estimation is taken into account. Similarly, the sampling period T is set from 0.003 s to 0.3 s with the step 0.009 s, and the estimation error curves of Algorithm 2 and Algorithm 1 in [32] with the truncation L = 35 are drawn in Fig. 9. Figure 9 presents the estimation errors via Algorithm 2 and Algorithm 1 in [32], respectively. It is similar to the conclusion in Example 1, and the sampling period also affects the accuracy of state estimation. However, the algorithm proposed in this paper is the better for the estimation accuracy with the appropriate sampling period.

Conclusion
This paper presents two types of FOCKFs to solve problem on the initial value influence. The proposed Algorithms 1 and 2 based on CKF and HECKF can both effectively reduce the influence of initial value on the state estimation, respectively. For a relatively small fractional-order α ∈ (0, 1), the more accurate state estimation is obtained by Algorithms 1 and 2 compared to Algorithm 1 in [32] based on CKF without the initial value compensation. But the CKF proposed in Algorithm 1 in [32] may be more suitable for larger fractional-orders α ∈ (0, 1) if the calculation time is taken into account. The future work is to consider the initial value problem in the adaptive fractional-order CKF with the unknown covariance matrices Q and R for nonlinear fractional-order systems.

Conflict of interest
The authors declare that they have no conflict of interest.