International Scholarly Research Notices

Research Article

Iterative Smooth Variable Structure Filter for Parameter Estimation

Table 1

Nomenclature.
SymbolCommentsSize
βˆ’ 𝟏 , + Notation denoting an inverse and a pseudoinverse, respectively
a ( b ) The b t h derivative of a
| | Absolute value
Μ‚ β€Œ Estimation value
𝐴 ∘ 𝐡 Schur product between 𝐴 and 𝐡
𝑇 Matrix transpose
𝐀 System matrix 𝑛 Γ— 𝑛
𝐁 Input matrix 𝑛 Γ— 1
Ξ” π‘Ž Difference between π‘Ž ’s actual and estimated values
𝐞 𝐳 Output’s estimation error vectors π‘š Γ— 1
𝐞 𝐱 State’s estimation error vectors 𝑛 Γ— 1
𝐸 ( a ) The expectation operator of the element a
𝜸 The SVSF’s coefficient matrix 𝑛 Γ— 𝑛
𝐇 Output matrix π‘š Γ— 𝑛
𝐈 𝑛 Γ— 𝑛 The identity matrix with dimensions of n Γ— n 𝑛 Γ— 𝑛
π‘˜ Time step value 1 Γ— 1
π‘˜ | π‘˜ A posteriori value
π‘˜ | π‘˜ βˆ’ 1 A priori value
𝐊 S V S F π‘˜ The SVSF’s gain 𝑛 Γ— 1
π‘š Number of measurements 1 Γ— 1
M a x E r r o r The maximum absolute error 1 Γ— 1
𝐌 π‘˜ A Lyapunov function 𝑛 Γ— 𝑛
𝑛 System’s number of states 1 Γ— 1
πœ” 𝑛 Natural frequency 1 Γ— 1
𝐏 Error covariance matrices 𝑛 Γ— 𝑛
Ξ¨ The smoothing boundary layer vector 𝑛 Γ— 1
𝐐 The process noise covariance matrix 𝑛 Γ— 𝑛
𝐑 The measurements noise covariance matrix π‘š Γ— π‘š
R M S E Root mean square error 1 Γ— 1
𝐬 𝐚 𝐭 ( 𝐚 , 𝐛 ) Saturation function of 𝐚 using the boundary layer 𝐛
s a t ( π‘Ž , 𝑏 ) Saturation function of element a using the boundary layer b 1 Γ— 1
βˆ‘ 𝑐 𝑖 = 𝑏 𝐚 𝑖 Summation of vector 𝐚 from time 𝑏 to time 𝑐
𝐬 𝐠 𝐧 ( 𝐚 ) Sign function of the vector 𝐚
s g n ( π‘Ž ) Sign function of the element a 1 Γ— 1
𝑇 𝑠 Sampling time 1 Γ— 1
𝑒 Input value 1 Γ— 1
𝐯 , 𝐕 m a x Measurement noise vector and its upper bound, respectively π‘š Γ— 1
𝐰 , 𝐖 m a x System noise vector and its upper bound, respectively 𝑛 Γ— 1
𝐱 State vector 𝑛 Γ— 1
𝐳 Output vector π‘š Γ— 1
𝟎 a Γ— b A matrix with dimension a Γ— b and zero elements π‘Ž Γ— 𝑏
πœ‰ Damping ratio 1 Γ— 1