PGopt is a software for determining optimal input trajectories with probabilistic performance and constraint satisfaction guarantees for unknown systems with latent states based on input-output measurements. In order to quantify uncertainties, which is crucial for deriving formal guarantees, a Bayesian approach is employed and a prior over the unknown dynamics and the system trajectory is formulated in state-space representation. Since for practical applicability, the prior must be updated based on input-output measurements, but the corresponding posterior distribution is analytically intractable, particle Gibbs (PG) sampling is utilized to draw samples from this distribution. Based on these samples, a scenario optimal control problem (OCP) is formulated and probabilistic performance and constraint satisfaction guarantees are inferred via a greedy constraint removal.

The approach is explained in the paper "Learning-Based Optimal Control with Performance Guarantees for Unknown Systems with Latent States", available as a preprint on arXiv.

Two versions of the algorithm are currently available: a Julia implementation and a MATLAB implementation.

In order to ensure the reproducibility of the results presented in the paper without reliance on proprietary software, a Julia implementation that utilizes the solver Altro to solve the OCP is provided. This version was used for the results presented in the paper and reproduces them exactly. However, this version has some limitations: only cost functions of the form $J_H=\sum\nolimits_{t=0}^H \frac{1}{2} u_t R u_t$, measurement functions of the form $y=x_{1:n_y}$, and output constraints of the form $y_\mathrm{min} \leq y \leq y_\mathrm{max}$ are supported.

Besides the Julia implementation that utilizes Altro, there is also an implementation that utilizes the solver IPOPT. This implementation allows arbitrary cost functions $J_H(u_{1:H},x_{1:H},y_{1:H})$, measurement functions $y=g(x,u)$, and constraints $h(u_{1:H},x_{1:H},y_{1:H})$. However, using IPOPT together with the proprietary HSL Linear Solvers for Julia (HSL_jll.jl) is recommended. A license (free to academics) is required.

Further information can be found in the PGopt Julia documentation.

The MATLAB implementation allows arbitrary cost functions $J_H(u_{1:H},x_{1:H},y_{1:H})$, measurement functions $y=g(x,u)$, and constraints $h(u_{1:H},x_{1:H},y_{1:H})$. CasADi and IPOPT are used to solve the scenario optimal control problem. In addition, the proprietary HSL Linear Solvers are used, which significantly accelerate the optimization. A license for the HSL Linear Solvers (free to academics) is required.

Further information can be found in the PGopt MATLAB documentation.

If you found this software useful for your research, consider citing us.

@article{PMCMC_OCP_arXiv_2023,
   title={Learning-Based Optimal Control with Performance Guarantees for Unknown Systems with Latent States},
   author={Lefringhausen, Robert and Srithasan, Supitsana and Lederer, Armin and Hirche, Sandra},
   journal={arXiv preprint arXiv:2303.17963},
   year={2023}
}