**1. Introduction**

Differing from the dynamic optimization problem (DOP), also known as the optimal control problem, in which only the control strategy decision is optimized to improve the performance of the dynamic system [1–3], the dynamic co-design and optimization (DCDO) problem accounts for the bi-directional dependency of physical system design and control system design and includes two types of design variables: plant (or physical) and control [4]. Two categories of co-design methods, nested (or multi-layer optimization) method and simultaneous method, are developed and deployed on the DCDO problem in the engineering applications [5–8]. In those two co-design methods, though the optimization structures are different, the DCDO problem should be transcribed into a finite-dimensional nonlinear programming (NLP) problem via direct transcription [9] to optimize.

However, when solving the DCDO problem of the sophisticated dynamic systems involving multiple disciplines or multiple subsystems, co-design schemes inevitably encounter obstructions such as high computational consumption incurred from the timeconsuming system simulations [10]. Moreover, in some engineering practices, dynamic system models are constructed by industrial simulation software or platforms, and the explicit equations of the state in dynamic systems expressed by differential algebraic equations cannot be extracted directly from the dynamic models [11]. The DCDO involving such dynamic system models is referred to as the black-box dynamic co-design and optimization

**Citation:** Zhang, Q.; Wu, Y.; Lu, L. A Novel Surrogate Model-Based Solving Framework for the Black-Box Dynamic Co-Design and Optimization Problem in the Dynamic System. *Mathematics* **2022**, *10*, 3239. https://doi.org/10.3390/ math10183239

Academic Editors: Camelia Petrescu and Valeriu David

Received: 19 August 2022 Accepted: 2 September 2022 Published: 6 September 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

(BDCDO) problem. Due to the lack of explicit state equation, the finite-difference technique, rather than the automatic differentiation technique, provides approximate Jacobian information matrices for NLP solvers to iteratively optimize BDCDO, which demands significant computational resources to evaluate the dynamic systems and definitely further increases the computational budget of the co-design schemes.

The response surface methods (RSMs) have been proven to be effective tools to address computationally expensive problems in complex static black-box systems [12–16]. Therefore, RSMs are introduced to approximate the black-box dynamic system to alleviate the number of the original system valuations and save computational costs. In attempting to reduce the modeling difficulty and preserve the dynamic properties of the dynamic system, Deshmukh et al. [17] presented the derivative function surrogate modeling methodology to construct surrogate models for the derivative functions of the dynamic system rather than construct surrogate models of the whole system responses [18–20]. In order to build high-fidelity model surrogate models of the derivative functions, Deshmukh et al. [17] used the Latin Hypercube Sampling (LHS) method for sequential sampling in the minimal hypercube space containing the current optimal trajectory to update the surrogate model. Lefebvre et al. [21] averaged the errors of the KRG model after integrating the errors along the current optimal trajectory and then used the values of state variables obtained by inverse error integral of each segment as new samples to update the KRG model. Qiao et al. [22] proposed EFDC sampling method based on KRG model to filter the current trajectory discrete points after error analysis, combine the spatial distance to cluster these discrete points, and select the points with the largest prediction error to update the model. At the same time, some reasonable solution termination criteria also have been investigated to avoid redundant iterations, which contribute little to the accuracy improvement of the solution result. Deshmukh et al. [17] determined whether the solution process stops or not based on the discrepancy between the current and previous iterates. Lefebvre et al. [21] proposed a new metric, dynamical mismatch, to identify whether the solution process is terminated or not. However, computing the dynamical mismatch needs additional sample points and consumes more computational resources. Qiao et al. [22] adopted the accuracy of the surrogate model and the successive relative improvement of the objective function as the stopping plan to assess the convergence of the solution process more comprehensively.

Admittedly, the BDCDO solving framework combined with the above-mentioned sampling strategies and termination criteria indeed reduce the number of samples for constructing the surrogate models of derivative functions to different degrees, but the efficiency of modeling and the robustness of the optimal solution still require further improvement. To this end, a new sequential sampling strategy based on the successive relative improvement ratio of the discrete trajectory points and maximizing distances between the nearest sample points, called SRIRMD, is proposed in this work to effectively improve the accuracy of the surrogate models by selecting the points with a large successive relative improvement ratio among the discrete trajectory points as new samples. Meanwhile, maximizing the minimum distances between new samples and existing samples can ensure the uniform distribution of all sample points. In addition, according to the fundamental observation that the state trajectories tend to coincide during the solving process of dynamic optimization problems, a new termination criterion, named the state trajectory overlap ratio (STOR), is presented to quantify the convergence and intuitively reflect the convergence trend of the solution in the iterative process. Finally, two numerical examples, one 3-DOF robot co-design and optimization problem and one horizontal axis wind turbine co-design and optimization problem, are solved by the means of the BDCDO solving framework integrated with the SRIRMD sampling strategy and the STOR termination criterion. The results demonstrate that the BDCDO solving framework combining the SRIRMD sampling strategy and the STOR termination criterion has the best performance compared to existing methods and can obtain more accurate and robust solutions with fewer sample points, improving the solution efficiency and reducing the computational budget.

The rest of this paper is organized as follows. Section 2 reviews the dynamic optimization problem and its direct solving method and the Kriging technique. Section 3 introduces the BDCDO solving framework combining the new sampling strategy and termination criterion. Section 4 verifies the feasibility and efficiency of the BDCDO solving framework through two numerical examples and two engineering examples. The conclusions are revealed in the last section.
