# Engineering Applications of Adaptive Kalman Filtering Based on Singular Value Decomposition (SVD)
## 3 Singular Value Decomposition  
Here, we consider singular value decomposition, which is a way of decomposing a matrix, A, in the product of three other matrices: \(A=USV^{T}\). **Remark 1**. : _Let \(A\) be an n\(\times\)d matrix with the right singular vectors \(v_{1}\), \(v_{2}\),..., \(v_{k}\): left singular vectors \(u_{1}\), \(u_{2}\),..., \(u_{k}\) ; and corresponding singular values \(\sigma_{1},\ \sigma_{2},\ \ldots,\sigma_{k}\). Then, \(A\) can be decomposed into a sum of rank-one matrices as follows:_  
\[A=\sum_{i=1}^{k}\sigma_{i}u_{i}v_{i}^{T}\approx\sigma_{1}u_{1}v_{1}^{T}+\sigma _{2}u_{2}v_{2}^{T}+\ldots+\sigma_{k}u_{k}v_{k}^{T}=U\Sigma V^{T}. \tag{10}\]  
_Here, \(U\in R^{m\times m}\) and \(V\in R^{n\times m}\), such that:_  
\[A=U\Sigma V^{T}, \tag{11}\]  
_where:_  
\[\Sigma=\left( \begin{array}{cc}S&0\\ 0&0\end{array} \right). \tag{12}\]  
_Here, \(\Sigma\in R^{m\times n}\) and \(S=\text{diag}(\sigma_{2},\ \ldots,\sigma_{r})\), with:_  
\[\sigma_{1}\geq\cdots\geq\ \sigma_{r}\geq 0,\]  
\[\sigma_{r+1}=0\cdots\sigma_{n}=0.,\]  
_The numbers \(\sigma_{1}\ldots\sigma_{r}\) along with \(\sigma_{r+1}=0,\ldots\sigma n=0\) are known as the singular values of \(A\), and they are the positive square roots of the eigenvalues of \(A^{T}A\)._  
Columns U of A are the orthonormal eigenvectors of \(AA^{T}\), whereas columns \(V\) of A are the orthonormal eigenvectors of \(A^{T}A\). It is understood that the singular values and singular vectors of a matrix are relatively insensitive to disturbance from the entries of the matrix and finite precision errors [8,20]. Given a non-square matrix \(A=U\ \Sigma V^{T}\), two matrices and their factorizations are of special interest:  
\[A^{T}\ A=V\Sigma^{2}V^{T}, \tag{13}\]  
\[AA^{T}\ =U\,\Sigma^{2}U^{T}. \tag{14}\]  
The algorithm is developed by comparing both sides of the obtained matrix equations. **4. SVD of Kalman Filter**  
In practical terms, if \(A^{T}\ A\) is positive definite, then Equation (13) can be compacted to:  
\[A=U\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\  
Equation (6) can then be written as:  
\[P_{k+1}^{-}=\Phi_{k+1}U_{k}^{+}D_{k}^{+2}U_{k}^{+T}\Phi_{k+1,k}^{T}+\Psi_{k}Q_{k} \,\Psi_{k}^{T}. \tag{18}\]  
The idea is to find the factors \(U_{k+1}\) and \(\mathrm{D}_{k+1}\) from Equation (18), such that:  
\[P_{k+1}=U_{k+1}D_{k+1}^{2}U_{k+1}^{T}. \tag{19}\]  
We define the following \(\left(\mathrm{s}+n\right)\times n\) matrix:  
\[\underbrace{\left( \begin{array}{c}D^{+2}U^{+T}\Phi_{k+1,k}^{T}\\ \left(\mathrm{Q}_{1}\frac{1}{2}\right)^{T}\Psi^{T}\end{array} \right)}_{Pre-array}. \tag{20}\]  
Then, we compute its singular value decomposition:  
\[\underbrace{\left( \begin{array}{c}D^{+2}U^{+T}\Phi_{k+1,k}^{T}\\ \left(\mathrm{Q}_{1}\frac{1}{2}\right)^{T}\Psi^{T}\end{array} \right)}_{Pre-array} =\underbrace{U_{k}^{T}\left( \begin{array}{c}D_{k}^{\prime}\\ 0\end{array} \right)V_{k}^{\prime}}_{Post-array}. \tag{21}\]  
Multiplying each term on the left by its transpose, we get:  
\[\Phi_{k+1,k}U_{k}^{+}D_{k}^{+T}D_{k}^{+}U_{k}^{+T}\Phi_{k+1,k}^{T}+\psi_{k}(Q _{k})^{\frac{1}{2}}\Big{(}Q_{k}^{\frac{1}{2}}\Big{)}^{T}\psi_{k}^{T}=V_{k}^{ \prime}\big{[}D_{k}^{\prime T}\big{]}\,0\big{]}U^{T}U\bigg{[} \begin{array}{c }D_{k}^{\prime}\\ 0\end{array} \bigg{]}V_{k}^{T}. \tag{22}\]  
We obtain:  
\[\Phi_{k+1,k}U_{k}^{+}D_{k}^{+2}\Phi_{k+1,k}^{T}+\psi_{k}Q_{k}\psi_{k}^{T}=V_{k }^{\prime}D_{k}^{\prime 2}V_{k}^{T}. \tag{23}\]  
Comparing the result of Equation (23) with Equations (18) and (19), it is easy to associate \(V_{k}^{\prime}\) and \(D_{k}^{\prime}\) with \(U_{k+1}\) and \(D_{k+1}\). In the orthodox Kalman measurement update, substituting Equation (8) into (9) yields:  
\[P_{k}^{+}=P_{k}^{-}-P_{k}^{-}H_{k}^{T}\Big{(}H_{k}P_{k}^{-}H_{k}^{T}+R_{k} \Big{)}^{-1}H_{k}P_{k}^{-}. \tag{24}\]  
**Lemma 1**.