γ-DDPC

γ-DDPC is a DDPC method presented in the following papers:

• Uncertainty-aware Data-Driven Predictive Control in a Stochastic Setting

• Data-Driven Predictive Control in a Stochastic Setting: A Unified Framework

Parameters and Variables

L is the LQ decomposition of the Hankel Matrix \([Z_p \ U_f \ Y_f]^T\).

\[\begin{split}L = \begin{bmatrix} L_{11} & 0 & 0 \\ L_{21} & L_{22} & 0 \\ L_{31} & L_{32} & L_{33} \end{bmatrix}\end{split}\]

Symbol	Dimensions
\(L\)	\((n\_z_p + n\_u_f + n\_y_f) \times (n\_z_p + n\_u_f + n\_y_f)\)
\(L_{11}\)	\(n\_z_p \times n\_z_p\)
\(L_{21}\)	\(n\_u_f \times n\_z_p\)
\(L_{22}\)	\(n\_u_f \times n\_u_f\)
\(L_{31}\)	\(n\_y_f \times n\_z_p\)
\(L_{32}\)	\(n\_y_f \times n\_u_f\)
\(L_{33}\)	\(n\_y_f \times n\_y_f\)

Optimization Formulation

\[\begin{split}\min_{u_f,y_f} &\quad \|y_f - y_r\|_Q^2 + \|u_f - u_r\|_R^2 + \lambda_{\gamma,2}||\gamma_2||_2^2 \\ \text{s.t.} &\quad u_f = L_{21} \gamma_1 + L_{22} \gamma_2 \\ &\quad y_f = L_{31} \gamma_1 + L_{32} \gamma_2 \\ &\quad \gamma_1^\star = L_{11}^{-1} z_p \\ &\quad u_f \in \mathcal{U}, \quad y_f \in \mathcal{Y}\end{split}\]

Closed-Form Solution Derivation

Solving for \(\gamma_2\)

\[\gamma_2 = -L_{22}^{-1}L_{21}L_{1}^{-1}z_p+L_{22}^{-1}u_f\]

Defining helper matrices for readability:

\[\begin{split}\begin{align*} T_1 &:= L_{22}^{-1}L_{21}L_{1}^{-1} \\ T_2 &:= L_{22}^{-1} \end{align*}\end{split}\]

Therefore:

\[\gamma_2 = -T_1z_p+T_2u_f\]

Replacing \(\gamma_1\), \(\gamma_2\):

\[y_f = (L_{31}L_{1}^{-1} - L_{32}T_1)z_p + L_{32}T_2 u_f\]

Defining helper matrices for readability:

\[\begin{split}\begin{align*} T_3 &:= (L_{31}L_{1}^{-1} - L_{32}T_1) \\ T_4 &:= L_{32}T_2 \end{align*}\end{split}\]

Therefore:

\[y_f = T_3z_p +T_4 u_f\]

The cost function can be written as:

\[\|T_3z_p+T_4u_f-y_r\|_Q^2 + \|u_f-u_r\|_R^2 + \lambda_{\gamma,2}\|-T_1z_p+T_2u_f\|^2_2\]

Taking the derivative with respect to \(u_f\) and setting it to zero leads to:

\[(T_4^TQT_4+R+\lambda_{\gamma,2}T_2^TT_2)u_f^\star = (T_2^TT_1-T_4^TT_3)z_p+T_4^TQy_r+Ru_r\]

Defining helper matrices:

\[\begin{split}\begin{align*} F_1 &= (T_4^TQT_4+R+\lambda_{\gamma,2}T_2^TT_2)^{-1} \\ F_2 &= (\lambda_{\gamma,2}T_2^TT_1-T_4^TQT_3) \\ F_3 &= T_4^TQ \end{align*}\end{split}\]

Leads to the following closed-form gain matrices:

\[\begin{split}K_{z_p} &= F_1 F_2 \\ K_{y_r} &= F_1 F_3 \\ K_{u_r} &= F_1 R\end{split}\]

With the optimal \(u_f^\star\) calculated as:

\[u_f^* = K_{z_p} z_p + K_{y_r} y_r + K_{u_r} u_r\]