Fernando A. C. C. Fontes Associate Professor
University of Porto Faculty of Engineering (FEUP) Department of Electrical and Computer
Engineering Rua Roberto Frias
4200-465 Porto Portugal faf@fe.up.pt web.fe.up.pt/∼faf
Mesh-Refinement Strategies for Fast Optimal Control and
Model Predictive Control of Kite Power Systems
Luís Tiago Paiva, Fernando A. C. C. Fontes
Research Center for Systems and Technologies (SYSTEC), Faculty of Engineering, University of Porto We consider continuous-time optimal control and model
predictive control problems for kite power systems. We solve these problems in a time-mesh that is adaptively re-fined to achieve a desired level of accuracy.
The proposed mesh-refinement strategy considers, in the optimal control problems (OCP) context, several refine-ment criteria in a multi-level scheme. A major feature of our strategy is to consider the local error of the dual vari-ables as a refinement criterion. This error can be com-puted efficiently by comparing the solution of a linear dif-ferential equation system – the adjoint equation of the maximum principle – with the numerically obtained mul-tipliers.
Details of this technique and its use in other nonlinear systems are reported in [1,2]. The refinement strategy resulted in higher accuracy levels and yet with lower overall computational time, when compared with the use of traditional meshes having equidistant-spacing in continuous-time OCP, and with a priori discretised ver-sions of the OCP. In particular, the time to obtain a solu-tion with a similar level of accuracy was 4% to 30% of the time needed using traditional equidistant meshes. The
benefits of using an adaptive time-mesh are particularly more evident in highly nonlinear systems, as is the case of the controlled kite and other non- holonomic systems. The technique led to the use of finer meshes in time inter-vals where sharp turns of the kite occur.
The refinement algorithm is extended to solve a sequence of optimal control problems in a Model Predictive Con-trol (MPC) scheme. In this extension, we consider a time-dependent stopping criterion for the mesh refinement al-gorithm with different levels. In particular, we impose a higher accuracy requirement in the initial parts of each horizon, which are more relevant in MPC. The use of adap-tive refinement in real-time optimization schemes, such as MPC, enables the possibility of obtaining a solution even when the optimization has to be interrupted at an early stage.
References:
[1] Paiva L. T., Fontes Fernando A. C. C.: Adaptive Time-Mesh Refine-ment in Optimal Control Problems with State Constraints. Discrete and Continuous Dynamical Systems, Vol. 35, No. 9 (2015) [2] Paiva L. T.: Numerical Methods in Optimal Control and Model Predictive Control. Ph.D. thesis, University of Porto, December 2014. http://hdl.handle.net/10216/77537
