Convex-Approximation-for-DeePC

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Convex-Approximation-for-DeePC

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# Regularization in DDPC: A Convex Relaxation Perspective
In this project, we introduce a novel bi-level optimization formulation based on the trajectory library of the dynamic system and discuss a series of its convex relaxation. This repository contains the MATLAB scripts for reproducing the experiments in our paper.

## Convex Relaxation for a Bi-level Formulation of Data-driven Predictive Control (DDPC)

Data-driven Predictive Control (*DDPC*) combines behavioral theory with receding horizon control has received increasing attention. It is first established for LTI systems and has been extended and applied for practical systems beyond LTI settings which has shown promising results. However, its mechanism for nondeterministic and nonlinear systems, and the relationship between different  *DDPC*  variants, involving regularization remains unclear.

In this paper, we introduce a new bi-level formulation incorporating both system ID techniques and predictive control, and discuss how existing and new variants of *DDPC* can be considered as convex relaxations of this bi-level formulation. Notably, a novel variant called *O-DDPC* has shown remarkable empirical performance on systems beyond deterministic LTI settings.
<figure>
 <img
 src="docs/img/comparison.png" align="center" width="100%" title="Schematic of data-driven control">
<figure>
Schematic of data-driven control

<figure>
 <img
 src="docs/img/Bilevel-formulation.png" align="center" width="60%" title="Bi-level Formulation">
<figure>
Bi-level Formulation

<figure>
 <img
 src="docs/img/Inner-problem.png" align="center" width="60%" title="Inner Problem">
<figure>
Inner Problem

## Dependency
The code requires the installation of [Mosek](https://www.mosek.com/) and the plot requires function [multiple_boxplot](https://www.mathworks.com/matlabcentral/fileexchange/47233-multiple_boxplot-m).

## Instruction
### Non-deterministic LTI System

- The `main_linear_Diff_Noise` can be used to reproduce results for non-deterministic LTI system with different pre-collected trajectories and different controllers. The pre-collected trajectories are in `Non_deterministic_LTI\data_100_New`. 
 - The results for reproducing Fig. 4 are in `results_Exp` and the figure is plotted by the `plot_avg_cost_var`. 
### Nonlinear System:
- The `main_Nonlinear` and can be used to reproduce results for systems with various of nonlinearity with different pre-collected trajectories. The pre-collected trajectories are in `Nonlinear\data_100_New`. 
- The results for reproducing Fig. 5 and Fig. 6 are in `results_Nonlinear`. The Fig. 5 and Fig. 6 are plotted by `plot_avg_cost` and `plot_avg_cost_var`, respectively.

# Contact us
To contact us about Regularization in DDPC, email either [Xu Shang](mailto:x3shang@ucsd.edu?Subject=Regularization-in-DDPC) or [Yang Zheng](mailto:zhengy@eng.ucsd.edu?Subject=Regularization-in-DDPC).

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