Optimization of dynamic engineering systems requires an integrated approach that accounts for the coupling between embodiment design and control system design, simultaneously. Generally known as combined design and control (co-design) optimization, these methods offer superior system's performance and reduced costs. Despite the widespread use of co-design approaches in the literature, not much work has been done to address the issue of uncertainty in co-design problem formulations. This is problematic as all engineering models contain some level of uncertainty that might negatively affect the system's performance, if overlooked. While in our previous study we developed a robust co-design approach, a more rigorous evaluation of probabilistic constraints is required to obtain the targeted reliability levels for probabilistic constraints. Therefore, we propose and implement a novel stochastic co-design approach based on the principles of reliability-based design optimization (RBDO) to explicitly account for uncertainties from design decision variables and problem parameters. In particular, a reliability-based, multidisciplinary dynamic system design optimization (RB-MDSDO) formulation is developed using the sequential optimization and reliability assessment (SORA) algorithm, such that the analysis-type dynamic equality constraints are satisfied at the mean values of random variables, as well as their most probable points (MPPs). The proposed approach is then implemented for two case studies and the results were benchmarked through Monte Carlo simulation (MCS) to indicate the impact of including reliability measures in co-design formulations.

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Optimization of dynamic engineering systems generally requires problem formulations that account for the coupling between embodiment design and control system design simultaneously. Such formulations are commonly known as combined optimal design and control (co-design) problems, and their application to deterministic systems is well established in the literature through a variety of methods. However, an issue that has not been addressed in the co-design literature is the impact of the inherent uncertainties within a dynamic system on its integrated design solution. Accounting for these uncertainties transforms the standard, deterministic co-design problem into a stochastic one, thus requiring appropriate stochastic optimization approaches for its solution. This paper serves as the starting point for research on stochastic co-design problems by proposing and solving a novel problem formulation based on robust design optimization (RDO) principles. Specifically, a co-design method known as multidisciplinary dynamic system design optimization (MDSDO) is used as the basis for an RDO problem formulation and implementation. The robust objective and inequality constraints are computed per usual as functions of their first-order-approximated means and variances, whereas analysis-based equality constraints are evaluated deterministically at the means of the random decision variables. The proposed stochastic co-design problem formulation is then implemented for two case studies, with the results indicating the importance of the robust approach on the integrated design solutions and performance measures.

Strong coupling of the physical and control parts within complex dynamic systems should be addressed by integrated design approaches that can manage such interactions. Otherwise, the final solution will be suboptimal or even infeasible. Combined design and control (co-design) methods can tackle this issue by managing the mentioned interactions and can result in superior optimal solutions. Current co-design methods are applicable to simplified non-interconnected systems; however, these methods might be impractical or even impossible to apply to real-world interconnected dynamic systems, hindering designers from obtaining the system-level optimal solutions. This work addresses this issue by developing an optimization algorithm which combines a decomposition-based optimization strategy known as analytical target cascading (ATC) with a co-design-centric formulation of multidisciplinary dynamic system design optimization (MDSDO). Considering the time-dependent linking variables among the dynamic systems’ components, a new consistency measure has also been proposed to manage such quantities in the optimization process. Finally, a plug-in hybrid electric vehicle powertrain, representative of an interconnected dynamic system, has been studied to validate the new algorithm’s results against the conventional all-at-once (AAO) MDSDO. Although the numerical results from the ATC-MDSDO slightly deviate from those in the AAO-MDSDO, this method can play a crucial role as a benchmark when the AAO solution is unattainable or a distributed design paradigm is required.

Combined plant and control design (control co-design, or CCD) methods are often used during product development to address the synergistic coupling between the plant and control parts of a dynamic system. Recently, a few studies have started applying CCD to stochastic dynamic systems. In their most rigorous approach, reliability-based design optimization (RBDO) principles have been used to ensure solution feasibility under uncertainty. However, since existing reliability-based CCD (RBCCD) algorithms use all-at-once (AAO) formulations, only most-probable-point (MPP) methods can be used as reliability analysis techniques. Though effective for linear/quadratic RBCCD problems, the use of such methods for highly nonlinear RBCCD problems introduces solution error that could lead to system failure. A multidisciplinary feasible (MDF) formulation for RBCCD problems would eliminate this issue by removing the dynamic equality constraints and instead enforcing them through forward simulation. Since the RBCCD problem structure would be similar to traditional RBDO problems, any of the well-established reliability analysis methods could be used. Therefore, in this work, a novel reliability-based MDF formulation of multidisciplinary dynamic system design optimization (RB-MDF-MDSDO) has been proposed for RBCCD. To quantify the uncertainty propagated by the random decision variables, Monte Carlo simulation has been applied to the generalized polynomial chaos (gPC) expansion of the probabilistic constraints. The proposed formulation is applied to two engineering test problems, with the results indicating the effectiveness of both the overall formulation as well as the reliability analysis technique for RBCCD.