Probabilistic analytical target cascading (PATC) has been developed to incorporate uncertainty of random variables in a hierarchical multilevel system using the framework of ATC. In the decomposed ATC structure, consistency between linked subsystems has to be guaranteed through individual subsystem optimizations employing special coordination strategies such as augmented Lagrangian coordination. However, the consistency in PATC has to be treated exploiting uncertainty quantification and propagation of interrelated linking variables that is the major concern of PATC. In previous works, the consistency of linking variables is assured by matching statistical moments under the normality assumption. However, it can induce significant error when the linking variable to be quantified is highly nonlinear and non-normal. In addition, reliability computed from statistical moments may be inaccurate in each optimization of the subsystem. To tackle the challenges, we propose the sampling-based PATC using kernel density estimation (KDE). The framework of reliability-based design optimization (RBDO) using sampling methods is adopted in individual optimizations of subsystems in the presence of uncertainty. The uncertainty quantification of a linking variable which is an intermediate random response can be achieved by shifted KDE. The constructed KDE based on finite samples of the linking variable can provide statistical representations to linked subsystems, and it can be utilized in the sampling-based RBDO through random samplings at the current design point. For the proposed sampling-based PATC, stochastic sensitivity analysis for KDE is further developed. The proposed sampling-based PATC using KDE facilitates efficient and accurate procedures to obtain a system optimum in PATC, and two examples based on mathematical function and finite element analysis (FEA) are used to demonstrate effectiveness of the proposed approach.
Analytical Target Cascading (ATC) is a decomposition-based optimization methodology that partitions a system into subsystems and then coordinates targets and responses among subsystems. Augmented Lagrangian with Alternating Direction method of multipliers (AL-AD), one of efficient ATC coordination methods, has been widely used in both hierarchical and non-hierarchical ATC and theoretically guarantees convergence under the assumption that all subsystem problems are convex and continuous. One of the main advantages of distributed coordination which consists of several non-hierarchical subproblems is that it can solve subsystem problems in parallel and thus reduce computational time. Therefore, previous studies have proposed an augmented Lagrangian coordination strategy for parallelization by eliminating interactions among subproblems. The parallelization is achieved by introducing a master problem and support variables or by approximating a quadratic penalty function to make subproblems separable. However, conventional AL-AD does not guarantee convergence in the case of parallel solving. Our study shows that, in parallel solving using targets and responses of the current iteration, conventional AL-AD causes mismatch of information in updating the Lagrange multiplier. Therefore, the Lagrange multiplier may not reach the optimal point, and as a result, increasing penalty weight causes numerical difficulty in the augmented Lagrangian coordination approach. To solve this problem, we propose a modified AL-AD with parallelization in non-hierarchical ATC. The proposed algorithm uses the subgradient method with adaptive step size in updating the Lagrange multiplier and also maintains penalty weight at an appropriate level not to cause oscillation. Without approximation or introduction of an artificial master problem, the modified AL-AD with parallelization can achieve similar accuracy and convergence with much less computational cost compared with conventional AL-AD with sequential solving.
Effective electrification of automotive vehicles requires designing the powertrain’s configuration along with sizing its components for a particular vehicle type. Employing planetary gear (PG) systems in hybrid electric vehicle (HEV) powertrain architectures allows various architecture alternatives to be explored, including single-mode architectures that are based on a fixed configuration and multimode architectures that allow switching power flow configuration during vehicle operation. Previous studies have addressed the configuration and sizing problems separately. However, the two problems are coupled and must be optimized together to achieve system optimality. An all-in-one (AIO) system solution approach to the combined problem is not viable due to the high complexity of the resulting optimization problem. This paper presents a partitioning and coordination strategy based on analytical target cascading (ATC) for simultaneous design of powertrain configuration and sizing for given vehicle applications. The capability of the proposed design framework is demonstrated by designing powertrains with one and two PGs for a midsize passenger vehicle.
This paper presents an industrial application of the analytical target cascading methodology to optimal design of commercial vehicle systems. The design problems concern the suspension of a heavy-duty truck and the body structure of a small bus. The results provide valuable insights in the feasibility of system-level design targets and the adequacy of subproblem design spaces during product development.
One approach to multiobjective optimization is to define a scalar substitute objective function that aggregates all objectives and solve the resulting aggregate optimization problem (AOP). In this paper, we discern that the objective function in quasi-separable multidisciplinary design optimization (MDO) problems can be viewed as an aggregate objective function (AOF). We consequently show that a method that can solve quasi-separable problems can also be used to obtain Pareto points of associated AOPs. This is useful when AOPs are too hard to solve or when the design engineer does not have access to the models necessary to evaluate all the terms of the AOF. In this case, decomposition-based design optimization methods can be useful to solve the AOP as a quasi-separable MDO problem. Specifically, we use the analytical target cascading methodology to formulate decomposed subproblems of quasi-separable MDO problems and coordinate their solution in order to obtain Pareto points of the associated AOPs.