Convex Approach to Data-Driven Optimal Control With Safety Constraints Using Linear Transfer Operator

Author

Joseph Raphel Moyalan, Clemson UniversityFollow

Date of Award

8-2024

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mechanical Engineering

Committee Chair/Advisor

Umesh Vaidya

Committee Member

Venkat Krovi

Committee Member

Phanindra Tallapragada

Committee Member

Ardalan Vahidi

Abstract

This thesis is concerned with the data-driven solution to the optimal control problem with safety constraints for a class of control-affine nonlinear systems. Designing optimal control satisfying safety constraints is a problem of interest in various applications, including robotics, power systems, transportation networks, and manufacturing. This problem is known to be non-convex. One of this thesis's main contributions is providing a convex formulation to this non-convex problem. The second main contribution is providing a data-driven framework for solving the control problem with safety constraints. The linear operator theoretic framework involving Perron-Frobenius and Koopman operators provides the convex formulation and associated data-driven computational framework for optimal control problems with safety constraints. In particular, the dual formulation of the optimal control problem in the space of density is convex.

This convex dual formulation of the optimal control problem is long well-known purely from the optimization perspective. However, the novelty of our approach is that we viewed the duality in the optimal control problem through the lenses of duality between the Koopman and Perron-Frobenius operators from the linear operator theory. This novel viewpoint has the following advantages. First, it provides a physical interpretation of the optimal cost in the dual space of density. The optimal density function and the measure associated with it have a physical interpretation of the occupancy of the system trajectories. The physical interpretation is exploited for the convex formulation of the safety constraints. Second, the approach has led to data-driven methods for optimal control computation. In particular, we show the dual convex, albeit infinite dimensional, formulation of the optimal control problem is closely related to the Perron-Frobenius operator. This has led to data-driven methods being developed to approximate the Koopman and Perron-Frobenius operators for the finite-dimensional data-driven approximation of the convex optimization problem. The computational framework relies on the polynomial basis and generator Extended Mode Dynamic Mode Decomposition (gEDMD) algorithm to approximate the Perron-Frobenius operator. The Sum of Square (SoS) optimization framework is utilized to impose positivity constraints in the optimization problem.

This thesis demonstrates the practical application of the convex optimal control formulation with safety constraints for the optimal navigation of a robotics system, specifically a quadruped robot, on unstructured terrain. The optimal navigation on the unstructured terrain extends the existing work on optimal navigation on flat terrain with a binary description of an unsafe set. The relative degree of traversability on the unstructured terrain is captured using a traversability map, leading to a probabilistic formulation of the navigation problem. The convex optimization framework is then applied to solve the probabilistic navigation problem, with a specific application to the navigation of a quadruped robot on unstructured terrain. Although convex, the optimization problem with safety constraints is computationally challenging to solve for systems with a large dimensional state space, as it involves approximating the Perron-Frobenius operator. To address this challenge, we introduce the 'control density function' (CDF) for the synthesis of a controller with safety constraints. The CDF can be viewed as a dual to the control barrier function (CBF), a popular approach to safe control design. While the safety certificate using the barrier function is based on the notion of invariance, the dual certificate involving the density function has a physical interpretation of occupancy. This occupancy-based physical interpretation is instrumental in providing an analytical construction of the density function used for safe control synthesis. The safe control design problem is formulated using the density function as a quadratic programming (QP) problem. In contrast to the QP proposed for control synthesis using CBF, the proposed CDF-based QP can combine the safety and convergence conditions to target state into single constraints. We also develop results for robust navigation using the CDF. We consider robustness against uncertainty in system dynamics and the initial condition. Finally, we present applications of CDF for robust navigation with a lane-keeping example, safe navigation in a double-gyre fluid flow field, and safe control design for a bicycle model. The robustness of our approach is one of the key strengths, ensuring the reliability and effectiveness of the control design in various scenarios.

Recommended Citation

Moyalan, Joseph Raphel, "Convex Approach to Data-Driven Optimal Control With Safety Constraints Using Linear Transfer Operator" (2024). All Dissertations. 3749.
https://open.clemson.edu/all_dissertations/3749

Author ORCID Identifier

0000-0002-0847-365X