TPC#
Transient Predictive Control (TPC) is a DDPC method with the following key properties:
Small online optimization problem,
Predictors are causal, and
not biased when closed-loop training data is used.
TPC is presented in the following research papers:
Parameters and Variables#
Symbol |
Definition |
Dimensions |
|---|---|---|
\(H_p,\ H_u\) |
Multistep predictor |
\(n\_y_f \times n\_z_p\); \(n\_y_f \times n\_u_f\) |
\(\Lambda_{uu},\ \Lambda_{uz},\ \Lambda_{uy}\) |
Regularization matrices |
\(n\_u_f \times n\_u_f\); \(n\_u_f \times n\_z_p\); \(n\_u_f \times n\_y_f\) |
Optimization Formulation#
\[\begin{split}\min_{u_f,y_f} &\quad \|y_f - y_r\|_Q^2 + \|u_f - u_r\|_R^2 + r(u_f, y_r, z_p) \\
\text{s.t.} &\quad y_f = H_u u_f + H_p z_p \\
&\quad u_f \in \mathcal{U}, \quad y_f \in \mathcal{Y}\end{split}\]
where:
\[r(u_f, y_r, z_p) := u_f^T \Lambda_{uu}u_f+2u^T_f \Lambda_{uz}z_p+2u_f^T \Lambda_{uy}y_r\]
Closed-Form Solution Derivation#
Set \(y_f\) into the cost equation leads to:
\[\|H_u u_f + H_p z_p - y_r\|_Q^2 + \|u_f - u_r\|_R^2 + u_f^T \Lambda_{uu}u_f+2u^T_f \Lambda_{uz}z_p+2u_f^T \Lambda_{uy}y_r\]
Now taking the derivative in respect to \(u_f\) and set it to zero.
\[(H_u ^T Q H_u + R + \Lambda_{uu})u_f^\star = (-H_u ^TQH_p -\Lambda_{uz})z_p + (H_u^TQ - \Lambda_{uy})y_r + R u_r\]
Defining the help matrices:
\[\begin{split}F_1 &= (H_u ^T Q H_u + R + \Lambda_{uu})^{-1} \\
F_2 &= -H_u ^TQH_p -\Lambda_{uz} \\
F_3 &= H_u^TQ - \Lambda_{uy}\end{split}\]
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:
\[u_f^* = K_{z_p} z_p + K_{y_r} y_r + K_{u_r} u_r\]