The paper “The Indirect Bang-Singular Algorithm (IBSA) for Singular Optimal Control Problems (OCPs) with State-Inequality Constraints” by Amin Jafarimoghaddam and Manuel Soler presents a novel algorithm aimed at addressing complex singular OCPs. The IBSA offers a general systematic and symbolic computational framework, addressing singular OCPs with singular controls expressed in terms of complete or incomplete state-feedback functions, fixed or unspecified terminal time, multiple singular controls, and state-inequality (or equality) constraints. These capabilities make the IBSA particularly suitable for solving advanced problems such as those encountered in commercial aircraft trajectory optimization.
Amin Jafarimoghaddam, Manuel Soler. The Indirect Bang-Singular Algorithm (IBSA) for singular control problems with state-inequality constraints. European Journal of Control, Volume 79, 2024, 101071, ISSN 0947-3580, https://doi.org/10.1016/j.ejcon.2024.101071.
Abstract
In this study, we present the Indirect Bang-Singular Algorithm (IBSA), a straightforward computational framework developed to solve a wide range of Singular Optimal Control Problems (SOCP) with state-inequality constraints. The algorithm is a type of arc-classification technique which reformulates the SOCP as a nonlinear programmingproblem over the switching times, and possibly some other parameters such as the co-states’ initial values at the entry time to a singular interval. We derive the singular control feedback using the Pontryagin’s maximum principle and analyze the possibility of an interval where multiple controls are simultaneously singular. Furthermore, we incorporate the state-inequality constraints using the direct-adjoining method. Owing to the linear property of the co-state dynamics, the co-state variables and consequently, the singular controls are computed automatically using MATLAB’s symbolic platform. The nonlinear programming is constructed in a manner to circumvent the challenges posed by state-inequality constraints in more intricate scenarios involving singular controls expressed in terms of incomplete state-feedback functions. We also present several theorems that are integral to devising a straightforward computational approach for solving SOCPs. To assess the effectiveness of the proposed algorithm, we solve the following novel problems: (1) time–fuel-optimal commercial aircraft cruise flight in a vertical plane (i.e., with state-inequality constraint, a scalar singular control, and wind shear effects), and (2) the free-routing time–fuel-optimal commercial aircraft flight in a vertical plane (i.e., with state-inequality constraint, a dual-entry singular control, and wind shear effects). Notably, the optimality of the graphed results has been carefully inspected through first and second-order optimality conditions.