Hybrid extended-unscented Kalman filters for continuous-time nonlinear fractional-order systems involving process and measurement noises

Problem statement

In this paper, we use two definitions of the fractional-order derivatives with respect to τ ( t ) , that is, Caputo definition in Machado et al. (2014) and GLD definition in Sierociuk and Dzielinski (2006).

By using the Caputo definition, we consider an estimated nonlinear system containing process and measurement noises to be investigated as follows

D t γ 0 C τ ( t ) = φ ( τ ( t ) , u ( t ) ) + w ( t ) ,

(1)

d ( t ) = η ( τ ( t ) ) + v ( t ) ,

(2)

where D t γ 0 C is the γ -order derivative with γ ∈ ( 0 , 2 ) . The state vector, the input and the measurement output are represented as τ ( t ) ∈ R n , u ( t ) ∈ R p and d ( t ) ∈ R q respectively, the nonlinear functions are represented by φ ( · , · ) and η ( · ) with C ∞ class, and w ( t ) ∈ R n and v ( t ) ∈ R q are the process and measurement noises respectively.

For the convenience of calculation, the sampling period T is denoted as the sampling time t = κ T . τ ( κ ) , u ( κ ) and d ( κ ) are defined as the sampling values of τ ( t ) , u ( t ) and d ( t ) . The mathematical expectations of w ( κ ) and v ( κ ) are 0 , and 0 is the zero matrix or vector with the appropriate dimension. Meanwhile, the covariance matrices of w ( κ ) and v ( κ ) are Q and R , and w ( κ ) and v ( κ ) are uncorrelated.

By using the GLD definition, the derivative D t γ 0 C τ ( t ) can be approximated by

D t γ 0 C τ ( t ) ≈ Δ γ τ ( κ ) = ( 1 T ) γ ∑ j = 0 κ ( − 1 ) j r j τ ( κ − j ) ,

(3)

where

r j = ( γ j ) = { 1 for j = 0 γ ( γ − 1 ) ⋯ ( γ − j + 1 ) j ! for j > 0 .

In order to gain more accurate estimation, we further investigate design KFs based on FOAD method for nonlinear systems with the uncorrelated noises. Within a sampling period T , the derivative D t γ 0 C τ ( t ) can be approximated using the FOAD method for t ∈ [ ( κ − 1 ) T , κ T ) as

D t γ 0 C τ ( t ) ≈ D t γ 0 C τ ( κ ) + 0 C D t γ τ ( κ − 1 ) 2 .

(4)

Therefore, the fractional-order nonlinear system (1) can be descretized by (4) as follows

( 1 T ) γ ∑ j = 0 κ ( − 1 ) j r j τ ( κ − j ) = φ ( τ ( κ ) , u ( κ ) ) + φ ( τ ( κ − 1 ) , u ( κ − 1 ) ) 2 + w ( κ ) + w ( κ − 1 ) 2 .

(5)

By choosing r ~ j = ( − 1 ) j 2 T γ r j , the equation (5) can be represented by

∑ j = 0 κ r ~ j τ ( κ − j ) = φ ( τ ( κ ) , u ( κ ) ) + φ ( τ ( κ − 1 ) , u ( κ − 1 ) ) + ∑ j = 0 1 w ( κ − j ) .

(6)

Compared with the UKF and EKF based on the GLD definition, the FOAD method can further strengthen the accuracy of state estimation. In the following section, HEUKFs for nonlinear fractional-order systems are proposed.

Design method of hybrid extended-unscented Kalman filters

In this subsection, the design method of the corresponding HEUKF is studied with the uncorrelated noises on the difference equation (6) derived by the GLD definition. According to the design method of extended Kalman filter, the calculation method of the first-order Taylor expansion is given for the nonlinear function at the current time.

The estimation of τ ( κ ) is defined as τ ^ ( κ | κ ) = E [ τ ( κ ) | λ ( κ ) ] , where λ ( κ ) involves d ( 0 ) , d ( 1 ) , . . . , d ( κ ) , u ( 0 ) , u ( 1 ) , . . . , u ( κ ) , then the first-order Taylor expression of the nonlinear function φ ( τ ( κ ) , u ( κ ) ) at τ ( κ ) = τ ^ ( κ − 1 | κ − 1 ) is given as follows

φ ( τ ( κ ) , u ( κ ) ) ≈ φ ( τ ^ ( κ − 1 | κ − 1 ) , u ( κ ) ) + ϕ ( κ − 1 ) ( τ ( κ ) − τ ^ ( κ − 1 | κ − 1 ) ) ,

(7)

where ϕ ( κ − 1 ) = ∂ φ ( τ , u ( κ ) ) ∂ τ | τ = τ ^ ( κ − 1 | κ − 1 ) .

Defining the nonlinear function ψ ( τ ( κ − 1 ) , u ( κ − 1 ) ) = φ ( τ ( κ − 1 ) , u ( κ − 1 ) ) − r ~ 1 τ ( κ − 1 ) and substituting (7) into (6), we have

[ r ~ 0 I − ϕ ( κ − 1 ) ] τ ( κ ) = ψ ( τ ( κ − 1 ) , u ( κ − 1 ) ) − ∑ j = 2 κ r ~ j τ ( κ − j ) + ∑ j = 0 1 w ( κ − j ) − ϕ ( κ − 1 ) τ ^ ( κ − 1 | κ − 1 ) + φ ( τ ^ ( κ − 1 | κ − 1 ) , u ( κ ) ) ,

(8)

where I is the identity matrix.

The nonlinear function ψ ~ ( τ ( κ − 1 ) , u ( κ − 1 ) ) and the matrices C ^ j ( κ − 1 ) , M ( κ − 1 ) are denoted for κ ≥ 1 as follows

ψ ~ ( τ ( κ − 1 ) , u ( κ − 1 ) ) = ( r ~ 0 I − ϕ ( κ − 1 ) ) − 1 ψ ( τ ( κ − 1 ) , u ( κ − 1 ) ) , C ^ j ( κ − 1 ) = − ( r ~ 0 I − ϕ ( κ − 1 ) ) − 1 r ~ j , M ( κ − 1 ) = ( r ~ 0 I − ϕ ( κ − 1 ) ) − 1 .

Meanwhile, M ( − 2 ) and M ( − 1 ) are determined by ϕ ( − 2 ) = ∂ φ ( τ , 0 ) ∂ τ | τ = 0 and ϕ ( − 1 ) = ∂ φ ( τ , u ( 0 ) ) ∂ τ | τ = 0 , which are used in the proposed HEUKFs.

Therefore, the difference equation (6) can be represented by

τ ( κ ) = ψ ~ ( τ ( κ − 1 ) , u ( κ − 1 ) ) + ∑ j = 2 κ C ^ j ( κ − 1 ) τ ( κ − j ) + ∑ j = 0 1 M ( κ − 1 ) w ( κ − j ) − M ( κ − 1 ) × [ ϕ ( κ − 1 ) τ ^ ( κ − 1 | κ − 1 ) − φ ( τ ^ ( κ − 1 | κ − 1 ) , u ( κ ) ) ] .

(9)

Then, the predication of τ ( κ ) is defined as τ ^ ( κ | κ − 1 ) = E [ τ ( κ ) | μ ( κ ) ] , where μ ( κ ) relates to d ( 0 ) , d ( 1 ) , …, d ( κ − 1 ) , u ( 0 ) , u ( 1 ) ,…, u ( κ ) . We assume that the condition E [ τ ( κ − j ) | μ ( κ ) ] ≅ E [ τ ( κ − j ) | λ ( κ − j ) ] can be obtained τ ^ ( κ | κ − 1 ) as follows

τ ^ ( κ | κ − 1 ) = E [ ψ ~ ( τ ( κ − 1 ) , u ( κ − 1 ) ) | μ ( κ ) ] + ∑ j = 2 κ C ^ j ( κ − 1 ) τ ^ ( κ − j | κ − j ) − M ( κ − 1 ) [ ϕ ( κ − 1 ) τ ^ ( κ − 1 | κ − 1 ) − φ ( τ ^ ( κ − 1 | κ − 1 ) , u ( κ ) ) ] .

(10)

For the uncorrelated noises w ( κ ) and v ( κ ) , the unscented Kalman filter is investigated. According to the viewpoint of unscented transformation (U-T), the calculation method of the σ points in terms of the nonlinear function Λ ( κ − 1 ) = ψ ~ ( τ ( κ − 1 ) , u ( κ − 1 ) ) is given as below

τ ( i ) ( κ − 1 | κ − 1 ) = { τ ^ ( κ − 1 | κ − 1 ) , i = 0 τ ^ ( κ − 1 | κ − 1 ) + ε i ( κ − 1 ) i = 1 , 2 , ⋯ , n τ ^ ( κ − 1 | κ − 1 ) − ε i − n ( κ − 1 ) i = n + 1 , ⋯ , 2 n ,

(11)

where ε i ( κ − 1 ) = ( ( n + ν ) P ( κ − 1 | κ − 1 ) ) i T , ( n + ν ) P ( κ − 1 | κ − 1 ) is the square root of the matrix ( n + ν ) P ( κ − 1 | κ − 1 ) , ( ( n + ν ) P ( κ − 1 | κ − 1 ) T ) i is the i th row of the matrix ( n + ν ) P ( κ − 1 | κ − 1 ) , ν = θ 2 ( n + ς ) − n , θ is a very small positive number, ς = 3 − n for white Gaussian noises and P ( κ − 1 | κ − 1 ) is the estimation error matrix for ( κ − 1 ) th iteration.

We investigate the estimation of τ ( κ ) = [ τ 1 ( κ ) , τ 2 ( κ ) , . . . , τ n ( κ ) ] T and δ ( κ ) = [ δ 1 ( κ ) , δ 2 ( κ ) , . . . , δ n ( κ ) ] T with τ k ( κ ) = τ ^ k ( κ | κ ) + δ τ k ( κ ) for k = 1 , 2 , . . . , n . By defining symbol ∇ = [ ∂ ∂ τ 1 ( κ − 1 ) , ∂ ∂ τ 2 ( κ − 1 ) , ⋯ , ∂ ∂ τ n ( κ − 1 ) ] T , the Taylor expansion for the nonlinear function Λ ( κ − 1 ) = ψ ~ ( τ ( κ − 1 ) , u ( κ − 1 ) ) with respect to τ ( κ − 1 ) at τ ( κ − 1 ) = τ ^ ( κ − 1 | κ − 1 ) is given by

Λ ( κ − 1 ) = ψ ~ ( τ ^ ( κ − 1 | κ − 1 ) , u ( κ − 1 ) ) + ∑ r = 1 + ∞ 1 r ! ( δ τ ( κ − 1 ) · ∇ ) r ψ ~ ( τ ^ ( κ − 1 | κ − 1 ) , u ( κ − 1 ) ) .

(12)

By denoting D δ τ ( κ − 1 ) r ψ ~ = ( δ τ ( κ − 1 ) · ∇ ) r ψ ~ ( τ ^ ( κ − 1 | κ − 1 ) , u ( κ − 1 ) ) , we obtain

Λ ( κ − 1 ) = ψ ~ ( τ ^ ( κ − 1 | κ − 1 ) , u ( κ − 1 ) ) + D δ τ ( κ − 1 ) ψ ~ + 1 2 ! D δ τ ( κ − 1 ) 2 ψ ~ + 1 3 ! D δ τ ( κ − 1 ) 3 ψ ~ + ⋯ .

(13)

For the function term 1 2 ! D δ τ ( κ − 1 ) 2 ψ ~ , we get

1 2 ! D δ τ ( κ − 1 ) 2 ψ ~ = 1 2 ( ∇ T δ τ ( κ − 1 ) δ T τ ( κ − 1 ) ∇ ) × ψ ~ ( τ ^ ( κ − 1 | κ − 1 ) , u ( κ − 1 ) ) .

(14)

We assume that τ ^ ( κ − 1 | κ − 1 ) is a mathematical expectation of the random variable τ ( κ − 1 ) . If the probability density is symmetric with respect to τ ^ ( κ − 1 | κ − 1 ) , we have E [ D δ τ ( κ − 1 ) 2 m − 1 ψ ~ ] = 0 for m = 1 , 2 , ⋯ .

From (14), the estimation of the nonlinear function Λ ( κ − 1 ) with the definition of P ( κ − 1 | κ − 1 ) as P ( κ − 1 | κ − 1 ) = E [ δ τ ( κ − 1 ) δ T τ ( κ − 1 ) ] can be represented by

Λ ^ ( κ − 1 | κ − 1 ) = ψ ~ ( τ ^ ( κ − 1 | κ − 1 ) , u ( κ − 1 ) ) + 1 2 ( ∇ T P ( κ − 1 | κ − 1 ) ∇ ) ψ ~ ( τ ^ ( κ − 1 | κ − 1 ) , u ( κ − 1 ) ) + O 1 .

(15)

where O 1 = E [ 1 4 ! D δ τ ( κ − 1 ) 4 ψ ~ + 1 6 ! D δ τ ( κ − 1 ) 6 ψ ~ + ⋯ ] .

From the σ points defined by (11), the nonlinear function Λ ( i ) ( κ − 1 | κ − 1 ) is yielded by τ ( i ) ( κ − 1 | κ − 1 ) . According to σ i ( κ − 1 ) = ε i ( κ − 1 ) for i = 1 , 2 , ⋯ , n and σ i ( κ − 1 ) = − ε i − n ( κ − 1 ) for i = n + 1 , n + 2 , ⋯ , 2 n , the Taylor expansion of Λ ( i ) ( κ − 1 | κ − 1 ) at τ ^ ( κ − 1 | κ − 1 ) for i = 1 , 2 , ⋯ , 2 n is determined by

Λ ( i ) ( κ − 1 | κ − 1 ) = ψ ~ ( τ ^ ( κ − 1 | κ − 1 ) , u ( κ − 1 ) ) + D σ i ( κ − 1 ) ψ ~ + 1 2 ! D σ i ( κ − 1 ) 2 ψ ~ + 1 3 ! D σ i ( κ − 1 ) 3 ψ ~ + ⋯ ,

(16)

where D σ i ( κ − 1 ) r ψ ~ = ( σ i ( κ − 1 ) · ∇ ) r ψ ~ ( τ ^ ( κ − 1 | κ − 1 ) , u ( κ − 1 ) ) .

For the case i = 0 , we know that τ ( 0 ) ( κ − 1 | κ − 1 ) = τ ^ ( κ − 1 | κ − 1 ) holds to obtain

Λ ( 0 ) ( κ − 1 | κ − 1 ) = ψ ~ ( τ ^ ( κ − 1 | κ − 1 ) , u ( κ − 1 ) ) .

(17)

We denote the following weights to estimate Λ ( κ − 1 ) using σ points as

W 0 ( m ) = ν n + ν ,

(18)

W i ( m ) = 1 2 ( n + ν ) , i = 1 , 2 , ⋯ , 2 n .

(19)

Accordingly, we get

Λ ^ U ( κ − 1 | κ − 1 ) = ∑ i = 0 2 n W i ( m ) Λ ( i ) ( κ − 1 | κ − 1 ) = ψ ~ ( τ ^ ( κ − 1 | κ − 1 ) , u ( κ − 1 ) ) + 1 2 ( n + ν ) × ∑ i = 1 2 n [ D σ i ( κ − 1 ) ψ ~ + 1 2 ! D σ i ( κ − 1 ) 2 ψ ~ + ⋯ ] .

(20)

Because the variable τ ( i ) ( κ − 1 | κ − 1 ) is symmetric with respect to mathematical expectation τ ^ ( κ − 1 | κ − 1 ) , we get ∑ i = 1 2 n D σ i ( κ − 1 ) 2 m − 1 ψ ~ = 0 for m = 1 , 2 , ⋯ , it follows that

Λ ^ U ( κ − 1 | κ − 1 ) = ψ ~ ( τ ^ ( κ − 1 | κ − 1 ) , u ( κ − 1 ) ) + 1 2 ( n + ν ) × ∑ i = 1 2 n [ 1 2 ! D σ i ( κ − 1 ) 2 ψ ~ + 1 4 ! D σ i ( κ − 1 ) 4 ψ ~ + ⋯ ] .

(21)

According to equation (11), we obtain

1 2 ( n + ν ) ∑ i = 1 2 n 1 2 D σ i ( κ − 1 ) 2 ψ ~ = 1 2 ( n + ν ) ∑ i = 1 2 n 1 2 ( ∇ T σ i ( κ − 1 ) σ i T ( κ − 1 ) ∇ ) ψ ~ ( τ ^ ( κ − 1 | κ − 1 ) , u ( κ − 1 ) ) = 1 2 ( ∇ T P ( κ − 1 | κ − 1 ) ∇ ) ψ ~ ( τ ^ ( κ − 1 | κ − 1 ) , u ( κ − 1 ) ) .

(22)

By using the condition (22), equation (21) can be rewritten as follows

Λ ^ U ( κ − 1 | κ − 1 ) = ψ ~ ( τ ^ ( κ − 1 | κ − 1 ) , u ( κ − 1 ) ) + 1 2 ( ∇ T P ( κ − 1 | κ − 1 ) ∇ ) ψ ~ ( τ ^ ( κ − 1 | κ − 1 ) , u ( κ − 1 ) ) + O 2 .

(23)

where O 2 = 1 2 ( n + ν ) E [ 1 4 ! D σ i ( κ − 1 ) 4 ψ ~ + 1 6 ! D σ i ( κ − 1 ) 6 ψ ~ + ⋯ ] .

From (15) and (23), the i th order terms with i = 0 , 1 , 2 , 3 in Λ ^ ( κ − 1 | κ − 1 ) and Λ ^ U ( κ − 1 | κ − 1 ) are similar to achieve the third-order approximations. If Λ ^ U ( κ − 1 | κ − 1 ) is used to replace Λ ^ ( κ − 1 | κ − 1 ) , a more satisfactory approximation of Λ ^ ( κ − 1 | κ − 1 ) in (10) is achieved.

Substituting (15) into (10), we obtain

τ ^ ( κ | κ − 1 ) = ψ ~ ( τ ^ ( κ − 1 | κ − 1 ) , u ( k − 1 ) ) + ∑ j = 2 κ C ^ j ( κ − 1 ) τ ^ ( κ − j | κ − j ) + 1 2 ( ∇ T P ( κ − 1 | κ − 1 ) ∇ ) ψ ~ ( τ ^ ( κ − 1 | κ − 1 ) , u ( κ − 1 ) ) − M ( κ − 1 ) × [ ϕ ( κ − 1 ) τ ^ ( κ − 1 | κ − 1 ) − φ ( τ ^ ( κ − 1 | κ − 1 ) , u ( κ ) ) ] + O 1 .

(24)

Replacing Λ ^ ( κ − 1 | κ − 1 ) with Λ ^ U ( κ − 1 | κ − 1 ) into (10), we also obtain

τ ^ U ( κ | κ − 1 ) = ∑ i = 0 2 n W i ( m ) Λ ( i ) ( κ − 1 | κ − 1 ) + ∑ j = 2 κ C ^ j ( κ − 1 ) τ ^ ( κ − j | κ − j ) − M ( κ − 1 ) [ ϕ ( κ − 1 ) τ ^ ( κ − 1 | κ − 1 ) − φ ( τ ^ ( κ − 1 | κ − 1 ) , u ( κ ) ) ] = ψ ~ ( τ ^ ( κ − 1 | κ − 1 ) , u ( κ − 1 ) ) + ∑ j = 2 κ C ^ j ( κ − 1 ) τ ^ ( κ − j | κ − j ) + 1 2 ( ∇ T P ( κ − 1 | κ − 1 ) ∇ ) ψ ~ ( τ ^ ( κ − 1 | κ − 1 ) , u ( κ − 1 ) ) − M ( κ − 1 ) [ ϕ ( κ − 1 ) τ ^ ( κ − 1 | κ − 1 ) − φ ( τ ^ ( κ − 1 | κ − 1 ) , u ( κ ) ) ] + O 2 .

(25)

Consequently, the three-order approximations with respect to δ τ ( κ − 1 ) can be achieved by replacing τ ^ ( κ | κ − 1 ) with τ ^ U ( κ | κ − 1 ) .

Approximations of state predication and predication error matrix

Substituting (13) into (9), the difference equation (9) can be represented by

τ ( κ ) = ψ ~ ( τ ^ ( κ − 1 | κ − 1 ) , u ( κ − 1 ) ) + D δ τ ( κ − 1 ) ψ ~ + 1 2 ! D δ τ ( κ − 1 ) 2 ψ ~ + ⋯ + ∑ j = 2 κ C ^ j ( κ − 1 ) τ ( κ − j ) − M ( κ − 1 ) [ ϕ ( κ − 1 ) τ ^ ( κ − 1 | κ − 1 ) − φ ( τ ^ ( κ − 1 | κ − 1 ) , u ( κ ) ) ] + ∑ j = 0 1 M ( κ − 1 ) w ( κ − j ) .

(26)

From (24) and (26), the predication error is calculated as follows

τ ( κ ) − τ ^ ( κ | κ − 1 ) = [ D δ τ ( κ − 1 ) ψ ~ + 1 2 D δ τ ( κ − 1 ) 2 ψ ~ + 1 3 ! D δ τ ( κ − 1 ) 3 ψ ~ + ⋯ ] − 1 2 ( ∇ T P ( κ − 1 | κ − 1 ) ∇ ) ψ ~ ( τ ^ ( κ − 1 | κ − 1 ) , u ( κ − 1 ) ) + ∑ j = 0 1 M ( κ − 1 ) w ( κ − j ) + ∑ j = 2 κ C ^ j ( κ − 1 ) δ ( κ − j ) − O 1 ,

(27)

where δ ( κ − j ) = τ ( κ − j ) − τ ^ ( κ − j | κ − j ) .

By neglecting higher-order terms, equation (27) can be approximated by

τ ( κ ) − τ ^ ( κ | κ − 1 ) ≈ D δ τ ( κ − 1 ) ψ ~ + ∑ j = 2 κ C ^ j ( κ − 1 ) δ ( κ − j ) + ∑ j = 0 1 M ( κ − 1 ) w ( κ − j ) .

(28)

Let P τ τ ( κ | κ − 1 ) = E [ ( τ ( κ ) − τ ^ ( κ | κ − 1 ) ) ( τ ( κ ) − τ ^ ( κ | κ − 1 ) ) T ] be the covariance matrix of estimation error by using (28), then P τ τ ( κ | κ − 1 ) is represented by

P τ τ ( κ | κ − 1 ) ≈ E [ ( D δ τ ( κ − 1 ) ψ ~ ) ( D δ τ ( κ − 1 ) ψ ~ ) T ] + E [ D δ τ ( κ − 1 ) ( ∑ j = 2 κ C ^ j ( κ − 1 ) δ ( κ − j ) ) T ] + E [ ( ∑ j = 2 κ C ^ j ( κ − 1 ) δ ( κ − j ) ) ( D δ τ ( κ − 1 ) ψ ~ ) T ] + E [ D δ τ ( κ − 1 ) ψ ~ ( ∑ j = 0 1 M ( κ − 1 ) w ( κ − j ) ) T ] + E [ ( ∑ j = 0 1 M ( κ − 1 ) w ( κ − j ) ) ( D δ τ ( κ − 1 ) ψ ~ ) T ]

+ E [ ( ∑ i = 2 κ C ^ i ( κ − 1 ) δ ( κ − i ) ) ( ∑ j = 2 κ C ^ j ( κ − 1 ) δ ( κ − j ) ) T ] + E [ ( ∑ i = 0 1 M ( κ − 1 ) w ( κ − i ) ) ( ∑ j = 0 1 M ( κ − 1 ) w ( κ − j ) ) T ] .

(29)

The estimation of the state vector is chosen as τ ^ ( κ | κ ) = τ ^ ( κ | κ − 1 ) + K ( κ ) ( d ( κ ) − η ( τ ^ ( κ | κ − 1 ) ) ) . Noting that d ( κ ) − η ( τ ^ ( κ | κ − 1 ) ) ≈ H ( κ ) ( τ ( κ ) − τ ^ ( κ | κ − 1 ) ) + v ( κ ) with H ( κ ) = ∂ η ( τ ) ∂ τ | τ = τ ^ ( κ | κ − 1 ) , the estimation error τ ^ ( κ | κ ) − τ ( κ ) by supposing that τ ^ ( κ − j | κ − j ) ≈ τ ( κ − j ) holds is calculated by

τ ^ ( κ | κ ) − τ ( κ ) ≈ ( I − K ( κ ) H ( κ ) ) ( − ∑ j = 0 1 M ( κ − 1 ) w ( κ − j ) ) + K ( κ ) v ( κ ) .

(30)

For the case i ≠ j , we obtain

E [ ( ∑ i = 2 κ C ^ i ( κ − 1 ) ( τ ^ ( κ − i | κ − i ) − τ ( κ − i ) ) ) × ( ∑ j = 2 κ C ^ j ( κ − 1 ) ( τ ^ ( κ − j | κ − j ) − τ ( κ − j ) ) ) T ] ≈ E [ ( ∑ i = 2 κ N i ( κ − 1 ) ( − ∑ s = 0 1 M i ( κ − 1 ) w ( κ − i − s ) ) + K ( κ − i ) v ( κ − i ) ) × ( ∑ j = 2 κ N j ( κ − 1 ) ( − ∑ s = 0 1 M j ( κ − 1 ) w ( κ − j − s ) ) + K ( κ − j ) v ( κ − j ) ) T ] = ∑ j = 2 κ − 1 N j ( κ − 1 ) M j ( κ − 1 ) QM j + 1 T ( κ − 1 ) N j + 1 T ( κ − 1 ) + ∑ j = 2 κ − 1 N j + 1 ( κ − 1 ) M j + 1 ( κ − 1 ) QM j T ( κ − 1 ) N j T ( κ − 1 ) ,

(31)

where N j ( κ − 1 ) = C ^ j ( κ − 1 ) ( I − K ( κ − j ) ) H ( κ − j ) and M j ( κ − 1 ) = M ( κ − j − 1 ) .

For the case i = j , we give

E [ ( ∑ i = 2 κ C ^ i ( κ − 1 ) δ ( κ − i ) ) ( ∑ j = 2 κ C ^ j ( κ − 1 ) δ ( κ − j ) ) T ] = ∑ j = 2 κ C ^ j ( κ − 1 ) P ( κ − j | κ − j ) C ^ j T ( κ − 1 ) .

(32)

Meanwhile, we have

E [ ( ∑ i = 2 κ C ^ i ( κ − 1 ) δ ( κ − i ) ) ( − ∑ j = 0 1 M ( κ − 1 ) w ( κ − j ) ) T ] = 0 .

(33)

Besides, we use equation (30) and replace κ with κ − 1 to obtain

τ ^ ( κ − 1 | κ − 1 ) − τ ( κ − 1 ) = − δ τ ( κ − 1 ) ≈ [ I − K ( κ − 1 ) H ( κ − 1 ) ] × [ − ∑ j = 1 2 M ( κ − 2 ) w ( κ − j ) ] + K ( κ − 1 ) v ( κ − 1 ) .

Based on the definition of D δ τ ( κ − 1 ) ψ ~ , we yield

D δ τ ( κ − 1 ) ψ ~ = ( ∇ ψ ~ ) T δ τ ( κ − 1 ) = − ( ∇ ψ ~ ) T [ ( I − K ( κ − 1 ) H ( κ − 1 ) ) × ( − ∑ j = 1 2 M ( κ − 2 ) w ( κ − j ) ) + K ( κ − 1 ) v ( κ − 1 ) ] .

(34)

From (34), we get

E [ D δ τ ( κ − 1 ) ψ ~ ( ∑ j = 2 κ C ^ j ( κ − 1 ) δ ( κ − j ) ) T ] = G T ( κ − 1 ) ( I − K ( κ − 1 ) H ( κ − 1 ) ) M ( κ − 2 ) Q M T ( κ − 3 ) × ( I − K ( κ − 2 ) H ( κ − 2 ) ) T C ^ 2 T ( κ − 2 )

(35)

and

E [ D δ τ ( κ − 1 ) ψ ~ ( ∑ j = 0 1 M ( κ − 1 ) w ( κ − j ) ) T ] = G T ( κ − 1 ) ( I − K ( κ − 1 ) H ( κ − 1 ) ) M ( κ − 2 ) Q M T ( κ − 1 ) ,

(36)

where G ( κ − 1 ) = ∂ ψ ~ ( τ , u ( κ − 1 ) ) ∂ τ | τ = τ ^ ( κ − 1 | κ − 1 ) .

It also has