Implicit integration with adjoint sensitivity propagation for optimal control problems involving differential-algebraic equations

Abstract

For the solution of optimal control problem involving an index-1 differential-algebraic equation, an efficient function evaluation algorithm is proposed in this paper. In the evaluation procedure, the state equation is propagated forwards, then, adjoint sensitivity is propagated backwards. Thus, it is computationally more efficient than forward sensitivity propagation when the number of constraints is less than that of optimization variables. In order to reduce Newton iterations, the adjoint sensitivity is derived utilizing the implicit function theorem, and the integration procedure is accelerated by incorporating a predictor-corrector strategy. This algorithm is integrated with a nonlinear programming solver Ipopt to solve sequentially the point-to-point optimal control for a Delta robot with constrained motor torque. Numerical experiments demonstrate the efficiency of this algorithm.

An efficient adjoint computational method based on lifted IRK integrator and exact penalty function for optimal control problems involving continuous inequality constraints

“…Thus, this method presents an efficient way to solve optimal control problems (OCPs) of middle or small scale without the assistance of commercial sparse linear algebraic algorithms. Then, this method is extended to sequential (or single-shooting) methods in [10,9], where corresponding forward and adjoint (backward) propagation algorithms are proposed for gradient evaluation. It is also shown in [10] that integration accuracy can be guaranteed by introducing constraints restricting the integration error.…”

“…This technique cannot guarantee the feasibility of solution in certain cases [8]. Moreover, when there are more constraints than optimization variables, the adjoint method proposed in [9] will be less efficient than the forward one [10]. Then, is there an efficient adjoint method to compute the optimal control subject to continuous inequality constraints?…”

“…Then, is there an efficient adjoint method to compute the optimal control subject to continuous inequality constraints? In this paper, the exact penalty function proposed in [8] is introduced in the adjoint method in [9], where continuous inequality constraints are transformed and penalized in the cost so that the surrogate problem has only box constraints on the variables. This penalty function is smooth and locally exact, which derives from [7] and has a form similar to those in [27,28,12,11,14,3].…”

“…Compared with [10], the proposed method is simplified, as time-scaling transformation [15] is not required to derive the sensitivity with respect to time steps. The proof of the adjoint sensitivity propagation rule is also provided in this paper, which serves as a complement to [9]. This computational method is based on the idea of control parametrization [13], and achieves improved efficiency by embedding a tailored integrator.…”

“…This computational method is based on the idea of control parametrization [13], and achieves improved efficiency by embedding a tailored integrator. It is demonstrated in [9] that the proposed IRK integrator enhanced by tangential prediction is superior to that in [17] when computational time is stringently constrained. As an improvement to method in [9], the computational method proposed in this paper provides an opportunity to solve OCPs with complex path constraints in real time.…”