Dynamic engineering systems are playing an increasingly important role in society, especially as active and autonomous dynamic systems become more mature and prevalent across a variety of domains. Successful design of complex dynamic systems requires multidisciplinary analysis and design techniques. While multidisciplinary design optimization (MDO) has been used successfully for the development of many dynamic systems, the established MDO formulations were developed around fundamentally static system models. We still lack general MDO approaches that address the specific needs of dynamic system design. In this article we review the use of MDO for dynamic system design, identify associated challenges, discuss related efforts such as optimal control, and present a vision for fully integrated design approaches. Finally, we lay out a set of exciting new directions that provide an opportunity for fundamental work in MDO. Nomenclature a(·)= analysis function a, b = example problem parameters α, β = energy domain designations A = state matrix for a linear and time invariant system B = input matrix for a linear and time invariant system c = suspension damping coefficient ε = convergence tolerance f (·)= design objective function f (·)= derivative function f a (·) = algebraic constraint g(·)= design constraint functions g p (·) = physical system constraints γ(t) = algebraic variable vector h i = time step i = time step index j = Gauss-Seidel block index, multiple-shooting time segment index k = iteration counter k s = suspension spring stiffness K = gain matrix K * = optimal gain matrix L(·) = Lagrange or running cost term m = number of Gauss-Seidel coordinate blocks n s = number of states n t = number of time steps n T = number of time segments φ(·) = cost function φ * (·) = optimal-value function (inner loop solution) φ(·) = alternative plant design objective function ψ(·) = Mayer or terminal cost term π(·) = augmented Lagrangian penalty function t = time t F = length of the time horizon t i = time at step i T j = time at the end of time segment j u(t) = control input trajectories u * (t) = optimal control trajectories u i = control input at time step i U = matrix discretization of u(t) x = optimization variable vector x * = optimal solution x k = solution estimate at iteration k x c = control system design variable vector x p = physical system design variable vector x p * = optimal plant design X = Cartesian product of closed convex sets ξ(t) = state variable trajectories ξ * (t) = optimal state trajectories ξ i = state at time step î ξ(t) = subset of state trajectorieṡ ξ(·) = time derivative of ξ(t) Ξ = discretization of ξ(t) Ξ = subset of discretized state trajectories y = coupling variable Y = matrix of initial state values for multiple shooting time segments ζ(·) = defect constraint functions (residuals) ζ i (·) = defect constraint between time segments
In this paper, general combined plant and control design or co-design problems are examined. The previous work in co-design theory imposed restrictions on the type of problems that could be posed. This paper lifts many of those restrictions. The problem formulations and optimality conditions for both the simultaneous and nested solution strategies are given. Due to a number of challenges associated with the optimality conditions, practical solution considerations are discussed with a focus on the motivating reasons for using direct transcription (DT) in co-design. This paper highlights some of the key concepts in general co-design including general coupling, the differences between the feasible regions for each strategy, general boundary conditions, inequality path constraints, system-level objectives, and the complexity of the closed-form solutions. Three co-design test problems are provided. A number of research directions are proposed to further co-design theory including tailored solution methods for reducing total computational expense, better comparisons between the two solution strategies, and more realistic test problems.
Energy extraction from ocean waves and conversion to electrical energy is a promising form of renewable energy, yet achieving economic viability of wave energy converters (WECs) has proven challenging. In this article, the design of a heaving cylinder WEC will be explored. The optimal plant (i.e. draft and radius) design space with respect to the design’s optimal control (i.e. power take-off trajectory) for maximum energy production is characterized. Irregular waves based on the Bretschneider wave spectrum are considered. The optimization problem was solved using a pseudospectral method, a direct optimal control approach that can incorporate practical design constraints, such as power flow, actuation force, and slamming. The results provide early-stage guidelines for WEC design. Results show the resonance frequency required for optimal energy production with a regular wave is quite different than the resonance frequency found for irregular waves; specifically, it is much higher.
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