The solution of complex system design problems using decomposition-based optimization methods requires determination of appropriate problem partitioning and coordination strategies. Previous optimal partitioning techniques have not addressed the coordination issue explicitly. This article presents a formal approach to simultaneous partitioning and coordination strategy decisions that can provide insights on whether a decomposition-based method will be effective for a given problem. Pareto-optimal solutions are generated to quantify tradeoffs between the sizes of subproblems and coordination problems as measures of the computational costs resulting from different partitioning and coordination strategies. Promising preliminary results with small test problems are presented. The approach is illustrated on an electric water pump design problem.

Design of physical systems and associated control systems are coupled tasks; design methods that manage this interaction explicitly can produce system-optimal designs, whereas conventional sequential processes may not. Here we explore a new technique for combined physical system and control design (co-design) based on a simultaneous dynamic optimization approach known as direct transcription, which transforms infinite-dimensional control design problems into finite dimensional nonlinear programming problems. While direct transcription problem dimension is often large, sparse problem structures and fine-grained parallelism (among other advantageous properties) can be exploited to yield computationally efficient implementations. Extension of direct transcription to co-design gives rise to a new problem structures and new challenges. Here we illustrate direct transcription for co-design using a new automotive active suspension design example developed specifically for testing co-design methods. This example builds on prior active suspension problems by incorporating a more realistic physical design component that includes independent design variables and a broad set of physical design constraints, while maintaining linearity of the associated differential equations.

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

Design of physical systems and associated control systems are coupled tasks; design methods that manage this interaction explicitly can produce system-optimal designs, whereas conventional sequential processes may not. Here, we explore a new technique for combined physical and control system design (co-design) based on a simultaneous dynamic optimization approach known as direct transcription, which transforms infinite-dimensional control design problems into finite-dimensional nonlinear programming problems. While direct transcription problem dimension is often large, sparse problem structures and fine-grained parallelism (among other advantageous properties) can be exploited to yield computationally efficient implementations. Extension of direct transcription to co-design gives rise to new problem structures and new challenges. Here, we illustrate direct transcription for co-design using a new automotive active suspension design example developed specifically for testing co-design methods. This example builds on prior active suspension problems by incorporating a more realistic physical design component that includes independent design variables and a broad set of physical design constraints, while maintaining linearity of the associated differential equations. A simultaneous co-design approach was implemented using direct transcription, and numerical results were compared with conventional sequential optimization. The simultaneous optimization approach achieves better performance than sequential design across a range of design studies. The dynamics of the active system were analyzed with varied level of control authority to investigate how dynamic systems should be designed differently when active control is introduced.

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