ASCC2022: GP+SOSP+Polynomial Controller

Consider a control affine dynamical system as follows, $$ \dot{x} = f(x)+g(x)u+d(x), $$ where $x \in \mathcal{X} \subset \mathbb{R}^{n}$ and $u \in \mathcal{U} \subset \mathbb{R}^{m}$ denote the state and control of the system. The system is consisted of three Lipschitz continuous terms, $f:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ denotes a nonlinear term, $g:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n\times m}$ denotes a polynomial term and $d:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ denotes an unknown term. We consider a polynomial control input $u$ over the stabilization process in this paper.

In this repo, we use

• Chebfun Toolbox: To approximate nonlinear terms by Chebyshev Interpolants,
• GPML Toolbox: Expressed the Gaussian processes mean function of this unknown term $d(x)$ into the polynomial form,
• SOSOPT+Mosek: To solve some sum-of-squares programmings in this learned polynomial system.

Note that, please run sosaddpath.m at the beginning and Do not forget to install the Mosek Solver in advance.

The final ROA with polynomial controller of the 2D system is:

The final ROA of the 3D demo:

The related files are concluded in the figure below

To verify the 2D demo, please run these files in a sequent.

• prepare_polynomial_system_1D.m
• demo_2d_lya_sublevelset.m
• demo_2d_opt_barrier.m
• demo_2d_Find_opt_Lya_original.m
• demo_2d_Find_opt_BV.m
• demo_2d_CLF_compare.m
• demo_2d_CLB_compare.m

To verify the 3D demo, please run these files in a sequent.

• prepare_polynomial_system_3D_2d.m
• demo_3d_lya_sublevelset.m
• demo_3d_opt_barrier.m
• demo_3d_Find_opt_Lya.m
• demo_3d_Find_opt_BV.m
• demo_3d_CLB_Comparer.m
• demo_3d_CLF_Comparer.m
• demo_3d_CLBF.m

Feel free to contact [email protected] for more details.