DeePC

Data-Enabled Predictive Control (DeePC) is a DDPC method presented in the following papers:

• Data-Enabled Predictive Control: In the Shallows of the DeePC

• Regularized and Distributionally Robust Data-Enabled Predictive Control

Performance Considerations

This method differs from other DDPC approaches by optimizing over a slack decision variable \(g\) of size \(n_{\text{col}}\). Since \(n_{\text{col}} \approx n_{\text{samples}}\) for large \(n_{\text{samples}}\), computational performance is directly impacted by \(n_{\text{samples}}\), leading to a computational complexity of \(\mathcal{O}(n_{\text{samples}}^2)\).

With a large number of samples, performance further degrades due to memory limitations.

Performance Benchmark (MacBook Pro, Double Integrator):

Number of Samples	Total Simulation Time
100	~1 second
300	~10 seconds
1000	~4 minutes

Note

100 samples are generally insufficient for the double integrator when process and measurement noise are present.

Noise Rejection

In the presence of noise, an additional slack variable \(\sigma\) is required to ensure feasibility of the equality constraints.

\[[Z_p] g = [z_p] + \sigma\]

where:

\[\begin{split}\sigma = \begin{bmatrix} \sigma_1 \\ 0 \\ \sigma_2 \\ \dots \\ \sigma_{\tau,f} \\ 0 \end{bmatrix}\end{split}\]

which is penalized by the cost function:

\[\lambda_{\sigma} \|\sigma\|_2^2\]

where \(\lambda_{\sigma}\) should be chosen large.

This allows the past measurements to deviate slightly from the Hankel matrices.

Additionally, a cost term is needed to regulate the size of \(g\) for a noise robust DeePC.

For further details, refer to Regularized and Distributionally Robust Data-Enabled Predictive Control.

Optimization Formulation

\[\begin{split} \min_{g, \sigma} &\quad [\|y_f - y_r\|_Q^2 + \|u_f - u_r\|_R^2 \\ &\quad + \lambda_{g,1} ||g||_1 + \lambda_{g,2} ||g||_2^2 \\ &\quad + \lambda_p ||(I-\Pi)g||_2^2 + \lambda_\sigma ||\sigma||_2^2 ]\\ \text{s.t.} &\quad \begin{bmatrix} Z_p \\ U_f \\ Y_f \end{bmatrix} g = \begin{bmatrix} z_p + \sigma \\ u_f \\ y_f \end{bmatrix}\\ &\quad u_f \in \mathcal{U}, \quad y_f \in \mathcal{Y}\end{split}\]

where \(\Pi\) is the Kernel Projection matrix calculated as follow:

\[\begin{split}\Pi &:= \begin{bmatrix} Z_p \\ U_f \end{bmatrix}^\dagger \begin{bmatrix} Z_p \\ U_f \end{bmatrix} \\ &= \begin{bmatrix} Z_p \\ U_f \end{bmatrix}^T \left( \begin{bmatrix} Z_p \\ U_f \end{bmatrix} \begin{bmatrix} Z_p \\ U_f \end{bmatrix}^T \right)^{-1} \begin{bmatrix} Z_p \\ U_f \end{bmatrix}\end{split}\]

Closed-Form Solution Derivation

Since the derivative of the 1-norm is not well-defined everywhere, \(\lambda_{g,1}\) must be set to zero for the derivation to be valid.

From the equality constraints, we obtain:

\[\begin{split}\sigma &= Z_p g - z_p \\ y_f &= Y_f g \\ u_f &= U_f g\end{split}\]

Substituting these into the cost function gives:

\[||Y_fg-y_r||_Q^2+ ||U_fg-u_r||^2_R + \lambda_{g,2}||g||_2^2 + \lambda_p||(I-\Pi)g||_2^2 + \lambda_\sigma||Z_pg-z_p||_2^2\]

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

\[(Y_f^TQY_f+U_f^TRU_f+\lambda_{g,2}+\lambda_p(I-\Pi)^T(I-\Pi)+\lambda_\sigma Z_p^T Z_p) g^\star = Y_f^TQy_r+U_f^TRu_r+\lambda_\sigma Z_p^T z_p\]

For readability:

\[\begin{split}T_1 &:= Y_f^TQY_f+U_f^TRU_f \\ T_2 &:=\lambda_{g,2}+\lambda_p(I-\Pi)^T(I-\Pi) \\ T_3 &:= \lambda_\sigma Z_p^T Z_p\end{split}\]

Replacing \(g^\star = U_f^{-1} u_f^\star\), we obtain the following helper matrices:

\[\begin{split}F_1 &:= U_f (T_1+T_2+T_3)^{-1} \\ F_2 &:= \lambda_\sigma Z_p^T \\ F_3 &:= Y_f^TQ \\ F_4 &:= U_f^TR\end{split}\]

This leads to the following closed-form gain matrices:

\[\begin{split}K_{z_p} &= F_1 F_2 \\ K_{y_r} &= F_1 F_3 \\ K_{u_r} &= F_1 F_4\end{split}\]

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

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