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 :math:`g` of size :math:`n_{\text{col}}`. Since :math:`n_{\text{col}} \approx n_{\text{samples}}` for large :math:`n_{\text{samples}}`, computational performance is directly impacted by :math:`n_{\text{samples}}`, leading to a computational complexity of :math:`\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): .. list-table:: :header-rows: 1 :widths: auto * - 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 :math:`\sigma` is required to ensure feasibility of the equality constraints. .. math:: [Z_p] g = [z_p] + \sigma where: .. math:: \sigma = \begin{bmatrix} \sigma_1 \\ 0 \\ \sigma_2 \\ \dots \\ \sigma_{\tau,f} \\ 0 \end{bmatrix} which is penalized by the cost function: .. math:: \lambda_{\sigma} \|\sigma\|_2^2 where :math:`\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 :math:`g` for a noise robust DeePC. For further details, refer to `Regularized and Distributionally Robust Data-Enabled Predictive Control `_. Optimization Formulation ------------------------ .. math:: \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} where :math:`\Pi` is the Kernel Projection matrix calculated as follow: .. math:: \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} Closed-Form Solution Derivation ------------------------------- Since the derivative of the 1-norm is not well-defined everywhere, :math:`\lambda_{g,1}` must be set to zero for the derivation to be valid. From the equality constraints, we obtain: .. math:: \sigma &= Z_p g - z_p \\ y_f &= Y_f g \\ u_f &= U_f g Substituting these into the cost function gives: .. math:: ||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 :math:`g` and setting it to zero: .. math:: (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: .. math:: 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 Replacing :math:`g^\star = U_f^{-1} u_f^\star`, we obtain the following helper matrices: .. math:: 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 This leads to the following closed-form gain matrices: .. math:: K_{z_p} &= F_1 F_2 \\ K_{y_r} &= F_1 F_3 \\ K_{u_r} &= F_1 F_4 With the optimal :math:`u_f^\star` calculated as: .. math:: u_f^* = K_{z_p} z_p + K_{y_r} y_r + K_{u_r} u_r