A Comparative Study of Sensitivity Computations in ESDIRK-Based Optimal Control Problems

In this paper, we compare the impact of iterated and direct approaches to sensitivity computation in fixed-step explicit singly diagonally-implicit Runge-Kutta (ESDIRK) methods when applied to optimal control problems (OCPs). We use the principle of internal numerical differentiation (IND) strictly for the iterated approach, i.e., reusing the iteration matrix factorizations, the number of Newton-type iterations, and Newton iterates, to compute the sensitivities. The direct method computes the sensitivities without using the Newton schemes. We compare the impact of the iterated and direct sensitivity computations in OCPs for the quadruple tank system. We benchmark the iterated and direct approaches with a base case. This base case is an OCP that applies an ESDIRK method that refactorizes the iteration matrix in every Newton iteration and uses a direct approach for sensitivity computations. In these OCPs, we vary the number of integration steps between control intervals and we evaluate the performance based on the number of SQP and QPs iterations, KKT violations, and the total number of function evaluations, Jacobian updates, and iteration matrix factorizations. The results indicate that the iterated approach outperforms the direct approach but yields similar performance to the base case.Comment: 6 pages, 5 figures, 2 tables. Submitted for European Control Conference 2024 (ECC2024). Stockholm, Swede

A progressive domain expansion method for solving optimal control problem

Electricity generation at the hydropower stations in Nigeria has been below the expected value. While the hydro stations have a capacity to generate up to 2,380 MW, the daily average energy generated in 2017 was estimated at around 846 MW. A factor responsible for this is the lack of a proper control system to manage the transfer of resources between the cascaded Kainji-Jebba Hydropower stations operating in tandem. This paper addressed the optimal regulation of the operating head of the Jebba hydropower reservoir in the presence of system constraints, flow requirement and environmental factors that are weather-related. The resulting two-point boundary value problem was solved using the progressive expansion of domain technique as against the shooting or multiple shooting techniques. The results provide the optimal inflow required to keep the operating head of the Jebba reservoir at a nominal level, hence ensuring that the maximum number of turbo-alternator units are operated