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Affine Residual Sum of Squares Gradient

Syntax
[grad] = ckbs_sumsq_grad(xzghdgdhqinvrinv)

Purpose
This computes the gradient of the of the affine Kalman-Bucy smoother residual sum of squares.

Notation
The affine Kalman-Bucy smoother residual sum of squares is defined by  \[
\begin{array}{rcl}
S ( x_1 , \ldots , x_N ) & = & \sum_{k=1}^N S_k ( x_k , x_{k-1} ) \\
S_k ( x_k , x_{k-1} )    & = & 
\frac{1}{2} 
( z_k - h_k - H_k * x_k )^\T * R_k^{-1} * ( z_k - h_k - H_k * x_k )
\\
& + &
\frac{1}{2} 
( x_k - g_k - G_k * x_{k-1} )^\T * Q_k^{-1} * ( x_k - g_k - G_k * x_{k-1} )
\end{array}
\] 
where the matrices  R_k and  Q_k are symmetric positive definite and  x_0 is the constant zero.

Gradient
We define  Q_{N+1} to be the  n \times n identity matrix and  G_{N+1} to be zero,  \[
\begin{array}{rcl}
\nabla_k S_k^{(1)} ( x_k , x_{k-1} ) 
& = &  H_k^\T * R_k^{-1} * ( h_k + H_k * x_k - z_k ) 
  +    Q_k^{-1} * ( x_k - g_k - G_k * x_{k-1} )
\\
\nabla_k S_{k+1}^{(1)} ( x_{k+1} , x_k ) 
& = & G_{k+1}^\T * Q_{k+1}^{-1} * ( g_{k+1} + G_{k+1} * x_k  - x_{k+1} )
\end{array}
\] 
It follows that the gradient of the affine Kalman-Bucy smoother residual sum of squares is  \[
\begin{array}{rcl}
\nabla S ( x_1 , \ldots , x_N ) 
& = &
\left( \begin{array}{c}
     d_1 \\ \vdots \\ d_N
\end{array} \right)
\\
d_k & = & \nabla_k S_k^{(1)}     ( x_k , x_{k-1} ) 
      +   \nabla_k S_{k+1}^{(1)} ( x_{k+1} , x_k )
\end{array}
\] 
where  S_{N+1} ( x_{N+1} , x_N ) is defined as identically zero.

x
The argument x is a two dimensional array, for  k = 1 , \ldots , N  \[
     x_k = x(:, k)
\]
and x has size  n \times N .

z
The argument z is a two dimensional array, for  k = 1 , \ldots , N  \[
     z_k = z(:, k)
\]
and z has size  m \times N .

g
The argument g is a two dimensional array, for  k = 1 , \ldots , N  \[
     g_k = g(:, k)
\]
and g has size  n \times N .

h
The argument h is a two dimensional array, for  k = 1 , \ldots , N  \[
     h_k = h(:, k)
\]
and h has size  m \times N .

dg
The argument dg is a three dimensional array, for  k = 1 , \ldots , N  \[
     G_k = dg(:,:,k)
\]
and dg has size  n \times n \times N .

dh
The argument dh is a three dimensional array, for  k = 1 , \ldots , N  \[
     H_k = dh(:,:,k)
\]
and dh has size  m \times n \times N .

qinv
The argument qinv is a three dimensional array, for  k = 1 , \ldots , N  \[
     Q_k^{-1} = qinv(:,:,k)
\]
and qinv has size  n \times n \times N .

rinv
The argument rinv is a three dimensional array, for  k = 1 , \ldots , N  \[
     R_k^{-1} = rinv(:,:,k)
\]
and rinv has size  m \times m \times N .

grad
The result grad is a two dimensional array, for  k = 1 , \ldots , N  \[
     d_k = grad(:, k)
\]
and grad has size  n \times N .

Example
The file sumsq_grad_ok.m contains an example and test of ckbs_sumsq_grad. It returns true if ckbs_sumsq_grad passes the test and false otherwise.
Input File: src/ckbs_sumsq_grad.m