An efficient direct/indirect transcription approach for singular optimal control

Abstract

It is very common in chemical engineering applications to find optimal control problems whose optimality conditions do not provide information about the control over an interval. This type of problems is called partially singular, as the control switches between nonsingular and singular arcs. When direct transcription is applied, the resulting nonlinear programming problem is ill conditioned. Some mesh refinement and rigorous iterative methods have been developed to determine the control profile and switching points. This work presents a practical alternative that quickly produces accurate state and control profiles without adding nonconvex terms. The problem is first solved with a large number of equally spaced finite elements. Then, unnecessary elements are removed while keeping the solution structure. Finally, direct and indirect approaches are combined to apply a regularization scheme only to the singular part. Seven examples were solved to test our strategy. Results provide good approximations to the analytical switching points.

The Switch Point Algorithm

“…CPU (s) Err ts 1 Our Study 0.20 1.56 × 10 −10 Nested Direct Transcription [9] 132.03 7.43 × 10 −06 Indirect Approach [2] 2.43 2.37 × 10 −09 Simultaneous Approach [10] 0.33 3.21 × 10 −07 Efficient Direct/Indirect Transcription [3] 0.39 3.76 × 10 −03…”

Solving singular control problems using uniform trigonometrization method

“…While solving the singular control problems with non-smooth control solutions using the PSM, additional steps are needed to obtain the singular control solution, which lead to inefficiencies and ill-conditioning of the resulting NLP problem as the corresponding Hessian becomes singular. [19][20][21][22] For instance, one PSM method was found deficient in solving the complicated problem of maximizing energy obtained from the waves in Ref. 23, which involved singular controls.…”

“…Andrés‐Martínez et al proposed a direct/indirect transcription approach in Ref. 22 that combined both direct and indirect methods to solve singular control problems. The direct/indirect approach, although being efficient, utilizes a strategy similar to the one used in GPOPS‐II 18 .…”

“…For brevity, the direct/indirect transcription strategy and the indirect approach proposed in Refs. 22 and 26, respectively, have not been included in this study for results and comparison with the UTM. Both these methods, however, offer unique features that are highly desirable for solving challenging OCPs with singular control arcs.…”

A simultaneous approach for singular optimal control based on partial moving grid

“…Betts 22 performed mesh refinement by solving a sequence of optimal control problems with an equally spaced fixed mesh. A related, similar approach is in described by Andrés-Martínez et al 23 Although each subproblem is well-posed, determining the switching point with high accuracy may be slow. Chen et al 24 decomposed the moving finite elements based simultaneous approach into an inner optimization problem that deals with a fixed mesh, and an outer optimization problem that adjusts the finite elements.…”