Unified Scaling of Dynamic Optimization Design Formulations

In this article, we explore scaling in dynamic optimization with a particular focus on how to leverage scaling in design studies. Here scaling refers to the process of suitable change of variables and algebraic manipulations to arrive at equivalent forms. The necessary theory for scaling dynamic optimization formulations is presented and a number of motivating examples are shown. The presented method is particularly useful for combined physical-system and control-system design problems to better understand the relationships between the optimal plant and controller designs. In one of the examples, scaling is used to understand observed results from more complete, higherfidelity design study. The simpler scaled optimization problem and dimensionless variables provide a number of insights. Scaling can be used to help facilitate finding accurate, generalizable, and intuitive information. The unique structure of dynamic optimization suggests that scaling can be utilized in novel ways to provide better analysis and formulations more favorable for efficiently generating solutions. The mechanics of scaling are fairly straightforward but proper utilization of scaling is heavily reliant on the creativity and intuition of the designer. The combination of existing theory and novel examples provides a fresh perspective on this classical topic in the context of dynamic optimization design formulations. Nomenclature Acronyms DAE Differential-algebraic equations DO Dynamic optimization ODE Direct transcription NLP Nonlinear program ODE Ordinary differential equation SASA Strain-actuated solar array Notation ̄ Scaled quantity ̇ Time derivative ′ Scaled derivative ∗ Optimal quantity Variables α Linear term in a scaling law β Constant term in a scaling law C Path constraints Constraint tolerance er Relative error f State derivative function (dynamics) H Hamiltonian L Lagrange (running cost) term in Ψ λ Costates or Lagrange multipliers for ξ M Mayer (terminal cost) term in Ψ O Optimality conditions for P p Time-independent optimization variables P Optimization problem φ Boundary constraints ρ Problem parameters ξ States t Time continuum u Open-loop control variables x Optimization variables ν, μ Lagrange multipliers for φ, C Proceedings of the ASME 2017 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference IDETC/CIE 2017 August 6-9, 2017, Cleveland, Ohio, USA