Overview
This article introduces a numerically viable technique for solving a specific class of constrained, non-convex, continuous-time optimal control problems (OCPs) applicable to linear systems. The method utilizes a lossless convexification technique to transform the original non-convex OCP, which includes annular control magnitude constraints, into a convex problem. Subsequently, the control space is finitely parameterized using piecewise constant functions to establish an efficient numerical approach.
Research Context
Optimal control problems are frequently encountered in various engineering domains, particularly in aerial and aerospace applications. Many real-world scenarios involve linear systems operating under continuous-time constraints, which often lead to non-convex OCPs. These non-convex problems, especially those with annular control magnitude constraints, traditionally present significant computational challenges in finding exact solutions. The development of methods that can guarantee exact solutions while efficiently handling an uncountable family of constraints over a compact time interval is relevant to improving guidance and control tasks in these fields.
Approach
The proposed technique involves two primary steps. First, the original non-convex OCP, characterized by its continuous-time nature and annular control magnitude constraints, undergoes a process of lossless convexification. This transformation converts the difficult non-convex problem into an equivalent convex problem, which is generally more tractable computationally. Second, to facilitate a numerical solution, the control space is parameterized. This parametrization involves representing the control functions as piecewise constant functions. This finite parametrization contributes to the efficiency of the numerical approach.
Findings
- The technique establishes a method for solving a class of constrained, non-convex, continuous-time optimal control problems for linear systems.
- Lossless convexification is employed to translate the original non-convex OCP, which includes annular control magnitude constraints, into a convex problem.
- The control space is finitely parameterized using piecewise constant functions.
- This approach provides an efficient numerical method that guarantees exact solutions.
- The method ensures the satisfaction of an uncountable family of constraints over a compact time interval.
- The effectiveness of this approach was demonstrated through a spacecraft landing problem involving three degrees of freedom (3 DoF).
Why This Matters
The established numerical technique offers an approach for obtaining exact solutions to a specific class of constrained optimal control problems. Its application is relevant to aerospace guidance and control tasks, as demonstrated through a spacecraft landing problem, suggesting its utility in real-world scenarios involving linear systems with continuous-time constraints.